author | Scott Morrison <scott@tqft.net> |
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%!TEX root = ../blob1.tex |
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\def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} |
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\def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip} |
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\section{$n$-categories and their modules} |
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\label{sec:ncats} |
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\subsection{Definition of $n$-categories} |
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\label{ss:n-cat-def} |
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Before proceeding, we need more appropriate definitions of $n$-categories, |
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$A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
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(As is the case throughout this paper, by ``$n$-category" we mean some notion of |
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a ``weak" $n$-category with ``strong duality".) |
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The definitions presented below tie the categories more closely to the topology |
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and avoid combinatorial questions about, for example, the minimal sufficient |
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collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
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For examples of topological origin |
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(e.g.\ categories whose morphisms are maps into spaces or decorated balls), |
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it is easy to show that they |
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satisfy our axioms. |
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For examples of a more purely algebraic origin, one would typically need the combinatorial |
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results that we have avoided here. |
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\medskip |
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There are many existing definitions of $n$-categories, with various intended uses. |
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In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
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Generally, these sets are indexed by instances of a certain typical shape. |
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Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, and so on). |
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Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, |
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a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, |
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and so on. |
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(This allows for strict associativity.) |
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Still other definitions (see, for example, \cite{MR2094071}) |
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model the $k$-morphisms on more complicated combinatorial polyhedra. |
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For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball. |
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Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
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to the standard $k$-ball. |
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By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
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standard $k$-ball. |
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We {\it do not} assume that it is equipped with a |
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preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
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Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
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the boundary), we want a corresponding |
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bijection of sets $f:\cC(X)\to \cC(Y)$. |
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(This will imply ``strong duality", among other things.) Putting these together, we have |
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\begin{axiom}[Morphisms] |
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\label{axiom:morphisms} |
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For each $0 \le k \le n$, we have a functor $\cC_k$ from |
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the category of $k$-balls and |
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homeomorphisms to the category of sets and bijections. |
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\end{axiom} |
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(Note: We usually omit the subscript $k$.) |
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We are being deliberately vague about what flavor of $k$-balls |
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we are considering. |
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They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
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They could be topological or PL or smooth. |
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%\nn{need to check whether this makes much difference} |
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(If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
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to be fussier about corners and boundaries.) |
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For each flavor of manifold there is a corresponding flavor of $n$-category. |
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For simplicity, we will concentrate on the case of PL unoriented manifolds. |
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(The ambitious reader may want to keep in mind two other classes of balls. |
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The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
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This will be used below to describe the blob complex of a fiber bundle with |
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base space $Y$. |
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The second is balls equipped with a section of the tangent bundle, or the frame |
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bundle (i.e.\ framed balls), or more generally some flag bundle associated to the tangent bundle. |
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These can be used to define categories with less than the ``strong" duality we assume here, |
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though we will not develop that idea fully in this paper.) |
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Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
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of morphisms). |
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The 0-sphere is unusual among spheres in that it is disconnected. |
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Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
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(Actually, this is only true in the oriented case, with 1-morphisms parameterized |
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by {\it oriented} 1-balls.) |
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For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. |
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For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. |
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(sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. |
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We prefer to not make the distinction in the first place. |
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Instead, we will combine the domain and range into a single entity which we call the |
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boundary of a morphism. |
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Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
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In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
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$1\le k \le n$. |
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At first it might seem that we need another axiom for this, but in fact once we have |
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all the axioms in this subsection for $0$ through $k-1$ we can use a colimit |
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construction, as described in \S\ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
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to spheres (and any other manifolds): |
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\begin{lem} |
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\label{lem:spheres} |
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For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from |
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the category of $k{-}1$-spheres and |
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homeomorphisms to the category of sets and bijections. |
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\end{lem} |
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We postpone the proof of this result until after we've actually given all the axioms. |
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Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, |
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along with the data described in the other axioms at lower levels. |
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%In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
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\begin{axiom}[Boundaries]\label{nca-boundary} |
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For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
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These maps, for various $X$, comprise a natural transformation of functors. |
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\end{axiom} |
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(Note that the first ``$\bd$" above is part of the data for the category, |
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while the second is the ordinary boundary of manifolds.) |
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Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
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Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
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The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
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all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
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(e.g.\ vector spaces, or modules over some ring, or chain complexes), |
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\nn{actually, need both disj-union/sub and product/tensor-product; what's the name for this sort of cat?} |
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and all the structure maps of the $n$-category should be compatible with the auxiliary |
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category structure. |
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Note that this auxiliary structure is only in dimension $n$; |
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$\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
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\medskip |
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(In order to simplify the exposition we have concentrated on the case of |
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unoriented PL manifolds and avoided the question of what exactly we mean by |
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the boundary a manifold with extra structure, such as an oriented manifold. |
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In general, all manifolds of dimension less than $n$ should be equipped with the germ |
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of a thickening to dimension $n$, and this germ should carry whatever structure we have |
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on $n$-manifolds. |
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In addition, lower dimensional manifolds should be equipped with a framing |
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of their normal bundle in the thickening; the framing keeps track of which |
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side (iterated) bounded manifolds lie on. |
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For example, the boundary of an oriented $n$-ball |
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should be an $n{-}1$-sphere equipped with an orientation of its once stabilized tangent |
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bundle and a choice of direction in this bundle indicating |
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which side the $n$-ball lies on.) |
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\medskip |
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We have just argued that the boundary of a morphism has no preferred splitting into |
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domain and range, but the converse meets with our approval. |
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That is, given compatible domain and range, we should be able to combine them into |
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the full boundary of a morphism. |
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The following lemma will follow from the colimit construction used to define $\cl{\cC}_{k-1}$ |
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on spheres. |
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\begin{lem}[Boundary from domain and range] |
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\label{lem:domain-and-range} |
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Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
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$B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
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Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
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two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
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Then we have an injective map |
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\[ |
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\gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S) |
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\] |
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which is natural with respect to the actions of homeomorphisms. |
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(When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
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becomes a normal product.) |
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\end{lem} |
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\begin{figure}[!ht] |
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$$ |
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178 |
\begin{tikzpicture}[%every label/.style={green} |
333 | 179 |
] |
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180 |
\node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {}; |
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\node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {}; |
186 | 182 |
\draw (S) arc (-90:90:1); |
183 |
\draw (N) arc (90:270:1); |
|
184 |
\node[left] at (-1,1) {$B_1$}; |
|
185 |
\node[right] at (1,1) {$B_2$}; |
|
186 |
\end{tikzpicture} |
|
187 |
$$ |
|
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188 |
\caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
179 | 189 |
|
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190 |
Note that we insist on injectivity above. |
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191 |
The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. |
109 | 192 |
|
333 | 193 |
Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$. |
402 | 194 |
We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
109 | 195 |
|
195 | 196 |
If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
333 | 197 |
as above, then we define $\cC(X)_E = \bd^{-1}(\cl{\cC}(\bd X)_E)$. |
195 | 198 |
|
333 | 199 |
We will call the projection $\cl{\cC}(S)_E \to \cC(B_i)$ |
110 | 200 |
a {\it restriction} map and write $\res_{B_i}(a)$ |
333 | 201 |
(or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)_E$. |
195 | 202 |
More generally, we also include under the rubric ``restriction map" the |
203 |
the boundary maps of Axiom \ref{nca-boundary} above, |
|
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another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition |
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205 |
of restriction maps. |
195 | 206 |
In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$ |
207 |
($i = 1, 2$, notation from previous paragraph). |
|
208 |
These restriction maps can be thought of as |
|
209 |
domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
|
94 | 210 |
|
211 |
||
212 |
Next we consider composition of morphisms. |
|
213 |
For $n$-categories which lack strong duality, one usually considers |
|
214 |
$k$ different types of composition of $k$-morphisms, each associated to a different direction. |
|
215 |
(For example, vertical and horizontal composition of 2-morphisms.) |
|
216 |
In the presence of strong duality, these $k$ distinct compositions are subsumed into |
|
217 |
one general type of composition which can be in any ``direction". |
|
218 |
||
187 | 219 |
\begin{axiom}[Composition] |
220 |
Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
|
179 | 221 |
and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
103 | 222 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
94 | 223 |
Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
224 |
We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
|
225 |
Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
|
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226 |
We have a map |
94 | 227 |
\[ |
228 |
\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
|
229 |
\] |
|
230 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
231 |
to the intersection of the boundaries of $B$ and $B_i$. |
|
232 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
187 | 233 |
(For $k=n$, see below.) |
234 |
\end{axiom} |
|
94 | 235 |
|
179 | 236 |
\begin{figure}[!ht] |
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237 |
$$ |
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238 |
\begin{tikzpicture}[%every label/.style={green}, |
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239 |
x=1.5cm,y=1.5cm] |
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\node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; |
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241 |
\node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; |
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242 |
\draw (S) arc (-90:90:1); |
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243 |
\draw (N) arc (90:270:1); |
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\draw (N) -- (S); |
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245 |
\node[left] at (-1/4,1) {$B_1$}; |
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246 |
\node[right] at (1/4,1) {$B_2$}; |
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247 |
\node at (1/6,3/2) {$Y$}; |
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248 |
\end{tikzpicture} |
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249 |
$$ |
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250 |
\caption{From two balls to one ball.}\label{blah5}\end{figure} |
179 | 251 |
|
195 | 252 |
\begin{axiom}[Strict associativity] \label{nca-assoc} |
187 | 253 |
The composition (gluing) maps above are strictly associative. |
254 |
\end{axiom} |
|
102 | 255 |
|
423 | 256 |
\nn{should say this means $N$ at a time, not just 3 at a time} |
257 |
||
179 | 258 |
\begin{figure}[!ht] |
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259 |
$$\mathfig{.65}{ncat/strict-associativity}$$ |
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260 |
\caption{An example of strict associativity.}\label{blah6}\end{figure} |
179 | 261 |
|
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262 |
We'll use the notations $a\bullet b$ as well as $a \cup b$ for the glued together field $\gl_Y(a, b)$. |
110 | 263 |
In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
195 | 264 |
a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
265 |
%Compositions of boundary and restriction maps will also be called restriction maps. |
|
266 |
%For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
|
267 |
%restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
|
110 | 268 |
|
192 | 269 |
We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
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270 |
We will call elements of $\cC(B)_Y$ morphisms which are |
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|
271 |
``splittable along $Y$'' or ``transverse to $Y$''. |
192 | 272 |
We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
109 | 273 |
|
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274 |
More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls. |
193 | 275 |
Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
276 |
the smaller balls to $X$. |
|
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277 |
We say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$". |
193 | 278 |
In situations where the subdivision is notationally anonymous, we will write |
279 |
$\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
|
280 |
the unnamed subdivision. |
|
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|
281 |
If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$; |
193 | 282 |
this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
283 |
subdivision of $\bd X$ and no competing subdivision of $X$. |
|
192 | 284 |
|
285 |
The above two composition axioms are equivalent to the following one, |
|
102 | 286 |
which we state in slightly vague form. |
287 |
||
288 |
\xxpar{Multi-composition:} |
|
289 |
{Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
|
290 |
into small $k$-balls, there is a |
|
291 |
map from an appropriate subset (like a fibered product) |
|
193 | 292 |
of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$, |
95 | 293 |
and these various $m$-fold composition maps satisfy an |
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294 |
operad-type strict associativity condition (Figure \ref{fig:operad-composition}).} |
179 | 295 |
|
296 |
\begin{figure}[!ht] |
|
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297 |
$$\mathfig{.8}{ncat/operad-composition}$$ |
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|
298 |
\caption{Operad composition and associativity}\label{fig:operad-composition}\end{figure} |
95 | 299 |
|
300 |
The next axiom is related to identity morphisms, though that might not be immediately obvious. |
|
301 |
||
343 | 302 |
\begin{axiom}[Product (identity) morphisms, preliminary version] |
303 |
For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, |
|
304 |
usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
|
305 |
These maps must satisfy the following conditions. |
|
306 |
\begin{enumerate} |
|
307 |
\item |
|
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308 |
If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are homeomorphisms such that the diagram |
343 | 309 |
\[ \xymatrix{ |
310 |
X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\ |
|
311 |
X \ar[r]^{f} & X' |
|
312 |
} \] |
|
313 |
commutes, then we have |
|
314 |
\[ |
|
315 |
\tilde{f}(a\times D) = f(a)\times D' . |
|
316 |
\] |
|
317 |
\item |
|
318 |
Product morphisms are compatible with gluing (composition) in both factors: |
|
319 |
\[ |
|
320 |
(a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D |
|
321 |
\] |
|
322 |
and |
|
323 |
\[ |
|
324 |
(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
|
325 |
\] |
|
326 |
\item |
|
327 |
Product morphisms are associative: |
|
328 |
\[ |
|
329 |
(a\times D)\times D' = a\times (D\times D') . |
|
330 |
\] |
|
331 |
(Here we are implicitly using functoriality and the obvious homeomorphism |
|
332 |
$(X\times D)\times D' \to X\times(D\times D')$.) |
|
333 |
\item |
|
334 |
Product morphisms are compatible with restriction: |
|
335 |
\[ |
|
336 |
\res_{X\times E}(a\times D) = a\times E |
|
337 |
\] |
|
338 |
for $E\sub \bd D$ and $a\in \cC(X)$. |
|
339 |
\end{enumerate} |
|
340 |
\end{axiom} |
|
341 |
||
342 |
We will need to strengthen the above preliminary version of the axiom to allow |
|
343 |
for products which are ``pinched" in various ways along their boundary. |
|
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|
344 |
(See Figure \ref{pinched_prods}.) |
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|
345 |
\begin{figure}[t] |
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|
346 |
$$ |
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347 |
\begin{tikzpicture}[baseline=0] |
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348 |
\begin{scope} |
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|
349 |
\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
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350 |
\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
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351 |
\foreach \x in {0, 0.5, ..., 6} { |
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352 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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353 |
} |
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354 |
\end{scope} |
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355 |
\draw[blue,line width=1.5pt] (0,-3) -- (5.66,-3); |
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356 |
\draw[->,red,line width=2pt] (2.83,-1.5) -- (2.83,-2.5); |
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357 |
\end{tikzpicture} |
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|
358 |
\qquad \qquad |
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|
359 |
\begin{tikzpicture}[baseline=-0.15cm] |
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360 |
\begin{scope} |
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361 |
\path[clip] (0,1) arc (90:135:8 and 4) arc (-135:-90:8 and 4) -- cycle; |
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362 |
\draw[blue,line width=2pt] (0,1) arc (90:135:8 and 4) arc (-135:-90:8 and 4) -- cycle; |
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363 |
\foreach \x in {-6, -5.5, ..., 0} { |
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\draw[green!50!brown] (\x,-2) -- (\x,2); |
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} |
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366 |
\end{scope} |
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\draw[blue,line width=1.5pt] (-5.66,-3.15) -- (0,-3.15); |
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\draw[->,red,line width=2pt] (-2.83,-1.5) -- (-2.83,-2.5); |
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369 |
\end{tikzpicture} |
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|
370 |
$$ |
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|
371 |
\caption{Examples of pinched products}\label{pinched_prods} |
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372 |
\end{figure} |
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|
373 |
(The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs} |
344 | 374 |
where we construct a traditional category from a topological category.) |
343 | 375 |
Define a {\it pinched product} to be a map |
376 |
\[ |
|
377 |
\pi: E\to X |
|
378 |
\] |
|
344 | 379 |
such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled |
343 | 380 |
on a standard iterated degeneracy map |
381 |
\[ |
|
344 | 382 |
d: \Delta^{k+m}\to\Delta^k . |
343 | 383 |
\] |
384 |
(We thank Kevin Costello for suggesting this approach.) |
|
385 |
||
344 | 386 |
Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball, |
343 | 387 |
and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension |
344 | 388 |
$l \le m$, with $l$ depending on $x$. |
343 | 389 |
|
390 |
It is easy to see that a composition of pinched products is again a pinched product. |
|
391 |
||
392 |
A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction |
|
393 |
$\pi:E'\to \pi(E')$ is again a pinched product. |
|
394 |
A {union} of pinched products is a decomposition $E = \cup_i E_i$ |
|
395 |
such that each $E_i\sub E$ is a sub pinched product. |
|
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396 |
(See Figure \ref{pinched_prod_unions}.) |
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|
397 |
\begin{figure}[t] |
364
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398 |
$$ |
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399 |
\begin{tikzpicture}[baseline=0] |
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400 |
\begin{scope} |
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401 |
\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
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402 |
\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
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403 |
\draw[blue] (0,0) -- (5.66,0); |
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404 |
\foreach \x in {0, 0.5, ..., 6} { |
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405 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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406 |
} |
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407 |
\end{scope} |
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408 |
\end{tikzpicture} |
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409 |
\qquad |
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410 |
\begin{tikzpicture}[baseline=0] |
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411 |
\begin{scope} |
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412 |
\path[clip] (0,-1) rectangle (4,1); |
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413 |
\draw[blue,line width=2pt] (0,-1) rectangle (4,1); |
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414 |
\draw[blue] (0,0) -- (5,0); |
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415 |
\foreach \x in {0, 0.5, ..., 6} { |
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416 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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417 |
} |
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418 |
\end{scope} |
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419 |
\end{tikzpicture} |
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420 |
\qquad |
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421 |
\begin{tikzpicture}[baseline=0] |
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422 |
\begin{scope} |
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423 |
\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4); |
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424 |
\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4); |
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425 |
\draw[blue] (2.83,3) circle (3); |
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426 |
\foreach \x in {0, 0.5, ..., 6} { |
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427 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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428 |
} |
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429 |
\end{scope} |
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430 |
\end{tikzpicture} |
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431 |
$$ |
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432 |
$$ |
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433 |
\begin{tikzpicture}[baseline=0] |
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434 |
\begin{scope} |
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435 |
\path[clip] (0,-1) rectangle (4,1); |
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436 |
\draw[blue,line width=2pt] (0,-1) rectangle (4,1); |
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437 |
\draw[blue] (0,-1) -- (4,1); |
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438 |
\foreach \x in {0, 0.5, ..., 6} { |
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439 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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440 |
} |
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441 |
\end{scope} |
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442 |
\end{tikzpicture} |
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443 |
\qquad |
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444 |
\begin{tikzpicture}[baseline=0] |
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445 |
\begin{scope} |
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446 |
\path[clip] (0,-1) rectangle (5,1); |
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447 |
\draw[blue,line width=2pt] (0,-1) rectangle (5,1); |
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448 |
\draw[blue] (1,-1) .. controls (2,-1) and (3,1) .. (4,1); |
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449 |
\foreach \x in {0, 0.5, ..., 6} { |
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450 |
\draw[green!50!brown] (\x,-2) -- (\x,2); |
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451 |
} |
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452 |
\end{scope} |
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453 |
\end{tikzpicture} |
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454 |
$$ |
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455 |
\caption{Five examples of unions of pinched products}\label{pinched_prod_unions} |
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|
456 |
\end{figure} |
343 | 457 |
|
458 |
The product axiom will give a map $\pi^*:\cC(X)\to \cC(E)$ for each pinched product |
|
459 |
$\pi:E\to X$. |
|
344 | 460 |
Morphisms in the image of $\pi^*$ will be called product morphisms. |
343 | 461 |
Before stating the axiom, we illustrate it in our two motivating examples of $n$-categories. |
462 |
In the case where $\cC(X) = \{f: X\to T\}$, we define $\pi^*(f) = f\circ\pi$. |
|
344 | 463 |
In the case where $\cC(X)$ is the set of all labeled embedded cell complexes $K$ in $X$, |
464 |
define $\pi^*(K) = \pi\inv(K)$, with each codimension $i$ cell $\pi\inv(c)$ labeled by the |
|
465 |
same (traditional) $i$-morphism as the corresponding codimension $i$ cell $c$. |
|
343 | 466 |
|
467 |
||
468 |
\addtocounter{axiom}{-1} |
|
187 | 469 |
\begin{axiom}[Product (identity) morphisms] |
344 | 470 |
For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
471 |
there is a map $\pi^*:\cC(X)\to \cC(E)$. |
|
340
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|
472 |
These maps must satisfy the following conditions. |
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473 |
\begin{enumerate} |
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474 |
\item |
344 | 475 |
If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and |
476 |
if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
|
95 | 477 |
\[ \xymatrix{ |
344 | 478 |
E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
95 | 479 |
X \ar[r]^{f} & X' |
480 |
} \] |
|
109 | 481 |
commutes, then we have |
482 |
\[ |
|
344 | 483 |
\pi'^*\circ f = \tilde{f}\circ \pi^*. |
109 | 484 |
\] |
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485 |
\item |
344 | 486 |
Product morphisms are compatible with gluing (composition). |
487 |
Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
|
488 |
be pinched products with $E = E_1\cup E_2$. |
|
489 |
Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$. |
|
490 |
Then |
|
109 | 491 |
\[ |
344 | 492 |
\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
109 | 493 |
\] |
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494 |
\item |
344 | 495 |
Product morphisms are associative. |
423 | 496 |
If $\pi:E\to X$ and $\rho:D\to E$ are pinched products then |
109 | 497 |
\[ |
344 | 498 |
\rho^*\circ\pi^* = (\pi\circ\rho)^* . |
109 | 499 |
\] |
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500 |
\item |
344 | 501 |
Product morphisms are compatible with restriction. |
502 |
If we have a commutative diagram |
|
503 |
\[ \xymatrix{ |
|
504 |
D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ |
|
505 |
Y \ar@{^(->}[r] & X |
|
506 |
} \] |
|
507 |
such that $\rho$ and $\pi$ are pinched products, then |
|
110 | 508 |
\[ |
344 | 509 |
\res_D\circ\pi^* = \rho^*\circ\res_Y . |
110 | 510 |
\] |
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|
511 |
\end{enumerate} |
187 | 512 |
\end{axiom} |
95 | 513 |
|
343 | 514 |
|
515 |
\medskip |
|
128 | 516 |
|
95 | 517 |
All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
518 |
The last axiom (below), concerning actions of |
|
519 |
homeomorphisms in the top dimension $n$, distinguishes the two cases. |
|
520 |
||
521 |
We start with the plain $n$-category case. |
|
522 |
||
420 | 523 |
\begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
187 | 524 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
95 | 525 |
to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
187 | 526 |
Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. |
267 | 527 |
\end{axiom} |
96 | 528 |
|
174 | 529 |
This axiom needs to be strengthened to force product morphisms to act as the identity. |
103 | 530 |
Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
96 | 531 |
Let $J$ be a 1-ball (interval). |
532 |
We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
|
122 | 533 |
(Here we use the ``pinched" version of $Y\times J$. |
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|
534 |
\nn{do we need notation for this?}) |
96 | 535 |
We define a map |
536 |
\begin{eqnarray*} |
|
537 |
\psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
|
538 |
a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
|
539 |
\end{eqnarray*} |
|
142 | 540 |
(See Figure \ref{glue-collar}.) |
189 | 541 |
\begin{figure}[!ht] |
542 |
\begin{equation*} |
|
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543 |
\begin{tikzpicture} |
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544 |
\def\rad{1} |
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\def\srad{0.75} |
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546 |
\def\gap{4.5} |
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547 |
\foreach \i in {0, 1, 2} { |
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548 |
\node(\i) at ($\i*(\gap,0)$) [draw, circle through = {($\i*(\gap,0)+(\rad,0)$)}] {}; |
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549 |
\node(\i-small) at (\i.east) [circle through={($(\i.east)+(\srad,0)$)}] {}; |
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550 |
\foreach \n in {1,2} { |
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\fill (intersection \n of \i-small and \i) node(\i-intersection-\n) {} circle (2pt); |
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552 |
} |
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553 |
} |
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554 |
|
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555 |
\begin{scope}[decoration={brace,amplitude=10,aspect=0.5}] |
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\draw[decorate] (0-intersection-1.east) -- (0-intersection-2.east); |
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\end{scope} |
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\node[right=1mm] at (0.east) {$a$}; |
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\draw (1-small) circle (\srad); |
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\draw[blue] (1) -- ($(1)+(\rad,0)+(\theta:\srad)$); |
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} |
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\filldraw[fill=white] (1) circle (\rad); |
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\foreach \n in {1,2} { |
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\fill (intersection \n of 1-small and 1) circle (2pt); |
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} |
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\node[below] at (1-small.south) {$a \times J$}; |
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\begin{scope} |
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\path[clip] (2) circle (\rad); |
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\foreach \y in {1, 0.86, ..., -1} { |
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\draw[blue] ($(2)+(-1,\y) $)-- ($(2)+(1,\y)$); |
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} |
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\end{scope} |
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579 |
\end{tikzpicture} |
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580 |
\end{equation*} |
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\begin{equation*} |
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582 |
\xymatrix@C+2cm{\cC(X) \ar[r]^(0.45){\text{glue}} & \cC(X \cup \text{collar}) \ar[r]^(0.55){\text{homeo}} & \cC(X)} |
189 | 583 |
\end{equation*} |
584 |
||
585 |
\caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
|
415
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586 |
We call a map of this form a {\it collar map}. |
96 | 587 |
It can be thought of as the action of the inverse of |
415
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588 |
a map which projects a collar neighborhood of $Y$ onto $Y$, |
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589 |
or as the limit of homeomorphisms $X\to X$ which expand a very thin collar of $Y$ |
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to a larger collar. |
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We call the equivalence relation generated by collar maps and homeomorphisms |
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isotopic (rel boundary) to the identity {\it extended isotopy}. |
96 | 593 |
|
594 |
The revised axiom is |
|
595 |
||
267 | 596 |
\addtocounter{axiom}{-1} |
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597 |
\begin{axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$.] |
187 | 598 |
\label{axiom:extended-isotopies} |
599 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
|
415
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600 |
to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
187 | 601 |
Then $f$ acts trivially on $\cC(X)$. |
415
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602 |
In addition, collar maps act trivially on $\cC(X)$. |
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\end{axiom} |
96 | 604 |
|
97 | 605 |
\smallskip |
606 |
||
607 |
For $A_\infty$ $n$-categories, we replace |
|
608 |
isotopy invariance with the requirement that families of homeomorphisms act. |
|
609 |
For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
|
416 | 610 |
Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
611 |
$C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
|
612 |
||
97 | 613 |
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614 |
\addtocounter{axiom}{-1} |
420 | 615 |
\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
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616 |
For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
97 | 617 |
\[ |
618 |
C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
|
619 |
\] |
|
620 |
These action maps are required to be associative up to homotopy |
|
621 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
|
236 | 622 |
a diagram like the one in Proposition \ref{CHprop} commutes. |
97 | 623 |
\nn{repeat diagram here?} |
187 | 624 |
\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
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625 |
\end{axiom} |
97 | 626 |
|
416 | 627 |
We should strengthen the above axiom to apply to families of collar maps. |
628 |
To do this we need to explain how collar maps form a topological space. |
|
629 |
Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
|
97 | 630 |
and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
416 | 631 |
Having chains on the space of collar maps act gives rise to coherence maps involving |
632 |
weak identities. |
|
420 | 633 |
We will not pursue this in detail here. |
97 | 634 |
|
103 | 635 |
Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
636 |
into a plain $n$-category (enriched over graded groups). |
|
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637 |
In a different direction, if we enrich over topological spaces instead of chain complexes, |
97 | 638 |
we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
639 |
instead of $C_*(\Homeo_\bd(X))$. |
|
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640 |
Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
97 | 641 |
type $A_\infty$ $n$-category. |
642 |
||
99 | 643 |
\medskip |
97 | 644 |
|
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645 |
The alert reader will have already noticed that our definition of a (plain) $n$-category |
416 | 646 |
is extremely similar to our definition of a system of fields. |
647 |
There are two differences. |
|
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648 |
First, for the $n$-category definition we restrict our attention to balls |
99 | 649 |
(and their boundaries), while for fields we consider all manifolds. |
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650 |
Second, in category definition we directly impose isotopy |
416 | 651 |
invariance in dimension $n$, while in the fields definition we |
652 |
instead remember a subspace of local relations which contain differences of isotopic fields. |
|
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653 |
(Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
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654 |
Thus a system of fields and local relations $(\cF,\cU)$ determines an $n$-category $\cC_ {\cF,\cU}$ simply by restricting our attention to |
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655 |
balls and, at level $n$, quotienting out by the local relations: |
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656 |
\begin{align*} |
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657 |
\cC_{\cF,\cU}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / \cU(B) & \text{when $k=n$.}\end{cases} |
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658 |
\end{align*} |
142 | 659 |
This $n$-category can be thought of as the local part of the fields. |
329
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660 |
Conversely, given a topological $n$-category we can construct a system of fields via |
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661 |
a colimit construction; see \S \ref{ss:ncat_fields} below. |
99 | 662 |
|
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\subsection{Examples of $n$-categories} |
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\label{ss:ncat-examples} |
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101 | 666 |
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667 |
We now describe several classes of examples of $n$-categories satisfying our axioms. |
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668 |
We typically specify only the morphisms; the rest of the data for the category |
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(restriction maps, gluing, product morphisms, action of homeomorphisms) is usually obvious. |
101 | 670 |
|
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671 |
\begin{example}[Maps to a space] |
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672 |
\rm |
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673 |
\label{ex:maps-to-a-space}% |
425 | 674 |
Let $T$ be a topological space. |
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We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. |
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For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of |
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all continuous maps from $X$ to $T$. |
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For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
196 | 679 |
homotopies fixed on $\bd X$. |
101 | 680 |
(Note that homotopy invariance implies isotopy invariance.) |
681 |
For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
|
682 |
be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
|
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683 |
\end{example} |
313 | 684 |
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685 |
\noop{ |
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Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above. |
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687 |
Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. |
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688 |
\nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from |
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689 |
an n-cat} |
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690 |
} |
101 | 691 |
|
423 | 692 |
\begin{example}[Maps to a space, with a fiber] \label{ex:maps-with-fiber} |
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693 |
\rm |
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\label{ex:maps-to-a-space-with-a-fiber}% |
196 | 695 |
We can modify the example above, by fixing a |
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closed $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, |
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697 |
otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. |
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698 |
Taking $F$ to be a point recovers the previous case. |
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699 |
\end{example} |
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700 |
|
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701 |
\begin{example}[Linearized, twisted, maps to a space] |
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702 |
\rm |
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703 |
\label{ex:linearized-maps-to-a-space}% |
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704 |
We can linearize Examples \ref{ex:maps-to-a-space} and \ref{ex:maps-to-a-space-with-a-fiber} as follows. |
101 | 705 |
Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$ |
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706 |
(have in mind the trivial cocycle). |
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707 |
For $X$ of dimension less than $n$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X)$ as before, ignoring $\alpha$. |
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708 |
For $X$ an $n$-ball and $c\in \Maps(\bdy X \times F \to T)$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X; c)$ to be |
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|
709 |
the $R$-module of finite linear combinations of continuous maps from $X\times F$ to $T$, |
101 | 710 |
modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
191
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711 |
$h: X\times F\times I \to T$, then $a = \alpha(h)b$. |
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712 |
(In order for this to be well-defined we must choose $\alpha$ to be zero on degenerate simplices. |
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713 |
Alternatively, we could equip the balls with fundamental classes.) |
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714 |
\end{example} |
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715 |
|
425 | 716 |
\begin{example}[$n$-categories from TQFTs] |
717 |
\rm |
|
718 |
\label{ex:ncats-from-tqfts}% |
|
719 |
Let $\cF$ be a TQFT in the sense of \S\ref{sec:fields}: an $n$-dimensional |
|
720 |
system of fields (also denoted $\cF$) and local relations. |
|
721 |
Let $W$ be an $n{-}j$-manifold. |
|
722 |
Define the $j$-category $\cF(W)$ as follows. |
|
723 |
If $X$ is a $k$-ball with $k<j$, let $\cF(W)(X) \deq \cF(W\times X)$. |
|
724 |
If $X$ is a $j$-ball and $c\in \cl{\cF(W)}(\bd X)$, |
|
725 |
let $\cF(W)(X; c) \deq A_\cF(W\times X; c)$. |
|
726 |
\end{example} |
|
727 |
||
728 |
The next example is only intended to be illustrative, as we don't specify |
|
729 |
which definition of a ``traditional $n$-category" we intend. |
|
730 |
Further, most of these definitions don't even have an agreed-upon notion of |
|
731 |
``strong duality", which we assume here. |
|
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732 |
\begin{example}[Traditional $n$-categories] |
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733 |
\rm |
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734 |
\label{ex:traditional-n-categories} |
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735 |
Given a ``traditional $n$-category with strong duality" $C$ |
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736 |
define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
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|
737 |
to be the set of all $C$-labeled embedded cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
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|
738 |
For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear |
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|
739 |
combinations of $C$-labeled embedded cell complexes of $X$ |
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740 |
modulo the kernel of the evaluation map. |
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|
741 |
Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
346
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|
742 |
with each cell labelled according to the corresponding cell for $a$. |
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|
743 |
(These two cells have the same codimension.) |
191
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744 |
More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$. |
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745 |
Define $\cC(X)$, for $\dim(X) < n$, |
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|
746 |
to be the set of all $C$-labeled embedded cell complexes of $X\times F$. |
191
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|
747 |
Define $\cC(X; c)$, for $X$ an $n$-ball, |
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|
748 |
to be the dual Hilbert space $A(X\times F; c)$. |
426
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|
749 |
(See \S\ref{sec:constructing-a-tqft}.) |
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|
750 |
\end{example} |
313 | 751 |
|
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|
752 |
\noop{ |
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|
753 |
\nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from |
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|
754 |
an n-cat} |
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|
755 |
Recall we described a system of fields and local relations based on a ``traditional $n$-category" |
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756 |
$C$ in Example \ref{ex:traditional-n-categories(fields)} above. |
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757 |
\nn{KW: We already refer to \S \ref{sec:fields} above} |
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|
758 |
Constructing a system of fields from $\cC$ recovers that example. |
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|
759 |
\todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} |
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|
760 |
\nn{KW: but the above example is all about string diagrams. the only difference is at the top level, |
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|
761 |
where the quotient is built in. |
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|
762 |
but (string diagrams)/(relations) is isomorphic to |
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|
763 |
(pasting diagrams composed of smaller string diagrams)/(relations)} |
418
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|
764 |
} |
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765 |
|
204 | 766 |
|
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|
767 |
\newcommand{\Bord}{\operatorname{Bord}} |
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|
768 |
\begin{example}[The bordism $n$-category, plain version] |
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|
769 |
\label{ex:bord-cat} |
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770 |
\rm |
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771 |
\label{ex:bordism-category} |
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772 |
For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
191
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|
773 |
submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
196 | 774 |
to $\bd X$. |
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|
775 |
For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such $n$-dimensional submanifolds; |
191
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|
776 |
we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
196 | 777 |
$W \to W'$ which restricts to the identity on the boundary. |
191
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|
778 |
\end{example} |
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|
779 |
|
196 | 780 |
%\nn{the next example might be an unnecessary distraction. consider deleting it.} |
101 | 781 |
|
196 | 782 |
%\begin{example}[Variation on the above examples] |
783 |
%We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
|
784 |
%for example product boundary conditions or take the union over all boundary conditions. |
|
785 |
%%\nn{maybe should not emphasize this case, since it's ``better" in some sense |
|
786 |
%%to think of these guys as affording a representation |
|
787 |
%%of the $n{+}1$-category associated to $\bd F$.} |
|
788 |
%\end{example} |
|
101 | 789 |
|
790 |
||
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|
791 |
%We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
101 | 792 |
|
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|
793 |
\begin{example}[Chains (or space) of maps to a space] |
191
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|
794 |
\rm |
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|
795 |
\label{ex:chains-of-maps-to-a-space} |
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|
796 |
We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$. |
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|
797 |
For a $k$-ball $X$, with $k < n$, the set $\pi^\infty_{\leq n}(T)(X)$ is just $\Maps(X \to T)$. |
191
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|
798 |
Define $\pi^\infty_{\leq n}(T)(X; c)$ for an $n$-ball $X$ and $c \in \pi^\infty_{\leq n}(T)(\bdy X)$ to be the chain complex |
418
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|
799 |
\[ |
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|
800 |
C_*(\Maps_c(X\times F \to T)), |
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|
801 |
\] |
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|
802 |
where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
101 | 803 |
and $C_*$ denotes singular chains. |
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|
804 |
Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X\times F \to T)$, |
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|
805 |
we get an $A_\infty$ $n$-category enriched over spaces. |
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|
806 |
\end{example} |
101 | 807 |
|
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|
808 |
See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
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|
809 |
homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
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|
810 |
|
279 | 811 |
\begin{example}[Blob complexes of balls (with a fiber)] |
812 |
\rm |
|
813 |
\label{ex:blob-complexes-of-balls} |
|
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|
814 |
Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
291 | 815 |
We will define an $A_\infty$ $k$-category $\cC$. |
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|
816 |
When $X$ is a $m$-ball, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
291 | 817 |
When $X$ is an $k$-ball, |
279 | 818 |
define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
819 |
where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
|
820 |
\end{example} |
|
101 | 821 |
|
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|
822 |
This example will be essential for Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
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823 |
Notice that with $F$ a point, the above example is a construction turning a topological |
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824 |
$n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. |
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|
825 |
We think of this as providing a ``free resolution" |
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|
826 |
of the topological $n$-category. |
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|
827 |
\nn{say something about cofibrant replacements?} |
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|
828 |
In fact, there is also a trivial, but mostly uninteresting, way to do this: |
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829 |
we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
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|
830 |
and take $\CD{B}$ to act trivially. |
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831 |
|
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|
832 |
Be careful that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
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|
833 |
It's easy to see that with $n=0$, the corresponding system of fields is just |
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|
834 |
linear combinations of connected components of $T$, and the local relations are trivial. |
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|
835 |
There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
191
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|
836 |
|
309
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|
837 |
\begin{example}[The bordism $n$-category, $A_\infty$ version] |
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|
838 |
\rm |
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|
839 |
\label{ex:bordism-category-ainf} |
348
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|
840 |
As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,\infty}(X)$ |
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|
841 |
to be the set of all $k$-dimensional |
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|
842 |
submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
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|
843 |
to $\bd X$. |
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|
844 |
For an $n$-ball $X$ with boundary condition $c$ |
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|
845 |
define $\Bord^{n,\infty}(X; c)$ to be the space of all $k$-dimensional |
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|
846 |
submanifolds $W$ of $X\times \Real^\infty$ such that |
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|
847 |
$W$ coincides with $c$ at $\bd X \times \Real^\infty$. |
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|
848 |
(The topology on this space is induced by ambient isotopy rel boundary. |
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|
849 |
This is homotopy equivalent to a disjoint union of copies $\mathrm{B}\!\Homeo(W')$, where |
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|
850 |
$W'$ runs though representatives of homeomorphism types of such manifolds.) |
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|
851 |
\nn{check this} |
309
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|
852 |
\end{example} |
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|
853 |
|
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|
854 |
|
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|
855 |
|
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|
856 |
Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little) |
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|
857 |
copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$. |
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|
858 |
(We require that the interiors of the little balls be disjoint, but their |
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|
859 |
boundaries are allowed to meet. |
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|
860 |
Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely |
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|
861 |
the embeddings of a ``little" ball with image all of the big ball $B^n$. |
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|
862 |
\nn{should we warn that the inclusion of this copy of $\Diff(B^n)$ is not a homotopy equivalence?}) |
419
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|
863 |
The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad: |
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|
864 |
by shrinking the little balls (precomposing them with dilations), |
346
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|
865 |
we see that both operads are homotopic to the space of $k$ framed points |
401
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|
866 |
in $B^n$. |
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|
867 |
It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have |
346
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|
868 |
an action of $\cE\cB_n$. |
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|
869 |
\nn{add citation for this operad if we can find one} |
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|
870 |
|
309
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|
871 |
\begin{example}[$E_n$ algebras] |
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|
872 |
\rm |
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|
873 |
\label{ex:e-n-alg} |
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|
874 |
|
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|
875 |
Let $A$ be an $\cE\cB_n$-algebra. |
346
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|
876 |
Note that this implies a $\Diff(B^n)$ action on $A$, |
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|
877 |
since $\cE\cB_n$ contains a copy of $\Diff(B^n)$. |
309
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|
878 |
We will define an $A_\infty$ $n$-category $\cC^A$. |
346
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|
879 |
If $X$ is a ball of dimension $k<n$, define $\cC^A(X)$ to be a point. |
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|
880 |
In other words, the $k$-morphisms are trivial for $k<n$. |
347
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|
881 |
If $X$ is an $n$-ball, we define $\cC^A(X)$ via a colimit construction. |
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|
882 |
(Plain colimit, not homotopy colimit.) |
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|
883 |
Let $J$ be the category whose objects are embeddings of a disjoint union of copies of |
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|
884 |
the standard ball $B^n$ into $X$, and who morphisms are given by engulfing some of the |
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|
885 |
embedded balls into a single larger embedded ball. |
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|
886 |
To each object of $J$ we associate $A^{\times m}$ (where $m$ is the number of balls), and |
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|
887 |
to each morphism of $J$ we associate a morphism coming from the $\cE\cB_n$ action on $A$. |
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|
888 |
Alternatively and more simply, we could define $\cC^A(X)$ to be |
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|
889 |
$\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$. |
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|
890 |
The remaining data for the $A_\infty$ $n$-category |
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|
891 |
--- composition and $\Diff(X\to X')$ action --- |
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|
892 |
also comes from the $\cE\cB_n$ action on $A$. |
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|
893 |
\nn{should we spell this out?} |
346
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|
894 |
|
424
6ebf92d2ccef
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|
895 |
\nn{Should remark that the associated hocolim for manifolds |
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|
896 |
is agrees with Lurie's topological chiral homology construction; maybe wait |
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|
897 |
until next subsection to say that?} |
356
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|
898 |
|
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remark about EB_n-algebras from n-cats
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|
899 |
Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
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|
900 |
$\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
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|
901 |
an $\cE\cB_n$-algebra. |
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|
902 |
\nn{The paper is already long; is it worth giving details here?} |
191
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|
903 |
\end{example} |
95 | 904 |
|
108 | 905 |
|
310
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|
906 |
\subsection{From balls to manifolds} |
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|
907 |
\label{ss:ncat_fields} \label{ss:ncat-coend} |
340
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|
908 |
In this section we describe how to extend an $n$-category $\cC$ as described above |
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|
909 |
(of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
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|
910 |
This extension is a certain colimit, and we've chosen the notation to remind you of this. |
402 | 911 |
Thus we show that functors $\cC_k$ satisfying the axioms above have a canonical extension |
340
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|
912 |
from $k$-balls to arbitrary $k$-manifolds. |
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|
913 |
Recall that we've already anticipated this construction in the previous section, |
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|
914 |
inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
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|
915 |
so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
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|
916 |
In the case of plain $n$-categories, this construction factors into a construction of a |
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|
917 |
system of fields and local relations, followed by the usual TQFT definition of a |
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|
918 |
vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
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|
919 |
For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
417
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changeset
|
920 |
Recall that we can take a plain $n$-category $\cC$ and pass to the ``free resolution", |
340
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|
921 |
an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls (recall Example \ref{ex:blob-complexes-of-balls} above). |
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|
922 |
We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
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|
923 |
for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the same as the original blob complex for $M$ with coefficients in $\cC$. |
108 | 924 |
|
417
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|
925 |
We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
340
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|
926 |
An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
419
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|
927 |
and we will define $\cl{\cC}(W)$ as a suitable colimit |
340
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|
928 |
(or homotopy colimit in the $A_\infty$ case) of this functor. |
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|
929 |
We'll later give a more explicit description of this colimit. |
420 | 930 |
In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data), |
340
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|
931 |
then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). |
108 | 932 |
|
419
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|
933 |
Define a {\it permissible decomposition} of $W$ to be a cell decomposition |
108 | 934 |
\[ |
935 |
W = \bigcup_a X_a , |
|
936 |
\] |
|
142 | 937 |
where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
419
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|
938 |
\nn{need to define this more carefully} |
108 | 939 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
191
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|
940 |
of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$. |
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|
941 |
|
419
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changeset
|
942 |
\begin{defn} |
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|
943 |
The category (poset) $\cell(W)$ has objects the permissible decompositions of $W$, |
340
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|
944 |
and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
191
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945 |
See Figure \ref{partofJfig} for an example. |
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|
946 |
\end{defn} |
119 | 947 |
|
948 |
\begin{figure}[!ht] |
|
949 |
\begin{equation*} |
|
222
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950 |
\mathfig{.63}{ncat/zz2} |
119 | 951 |
\end{equation*} |
329
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|
952 |
\caption{A small part of $\cell(W)$} |
119 | 953 |
\label{partofJfig} |
954 |
\end{figure} |
|
955 |
||
191
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956 |
An $n$-category $\cC$ determines |
329
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|
957 |
a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
108 | 958 |
(possibly with additional structure if $k=n$). |
197 | 959 |
Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
960 |
and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
|
961 |
are splittable along this decomposition. |
|
108 | 962 |
|
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963 |
\begin{defn} |
329
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|
964 |
Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
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|
965 |
For a decomposition $x = \bigcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
191
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|
966 |
\begin{equation} |
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967 |
\label{eq:psi-C} |
197 | 968 |
\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
191
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|
969 |
\end{equation} |
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970 |
where the restrictions to the various pieces of shared boundaries amongst the cells |
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971 |
$X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). |
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972 |
If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
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973 |
\end{defn} |
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974 |
|
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975 |
If $k=n$ in the above definition and we are enriching in some auxiliary category, |
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|
976 |
we need to say a bit more. |
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|
977 |
We can rewrite Equation \ref{eq:psi-C} as |
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978 |
\begin{equation} \label{eq:psi-CC} |
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|
979 |
\psi_{\cC;W}(x) \deq \coprod_\beta \prod_a \cC(X_a; \beta) , |
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980 |
\end{equation} |
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981 |
where $\beta$ runs through labelings of the $k{-}1$-skeleton of the decomposition |
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|
982 |
(which are compatible when restricted to the $k{-}2$-skeleton), and $\cC(X_a; \beta)$ |
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983 |
means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agress with $\beta$. |
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984 |
If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in |
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985 |
$\cS$ and the coproduct and product in Equation \ref{eq:psi-CC} should be replaced by the approriate |
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|
986 |
operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect). |
191
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987 |
|
420 | 988 |
Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
191
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989 |
|
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990 |
\begin{defn}[System of fields functor] |
415
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|
991 |
\label{def:colim-fields} |
402 | 992 |
If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
191
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993 |
That is, for each decomposition $x$ there is a map |
402 | 994 |
$\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps |
995 |
above, and $\cl{\cC}(W)$ is universal with respect to these properties. |
|
191
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996 |
\end{defn} |
112 | 997 |
|
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998 |
\begin{defn}[System of fields functor, $A_\infty$ case] |
402 | 999 |
When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$ |
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1000 |
is defined as above, as the colimit of $\psi_{\cC;W}$. |
402 | 1001 |
When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
191
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1002 |
\end{defn} |
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1003 |
|
402 | 1004 |
We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
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1005 |
with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
111 | 1006 |
|
422 | 1007 |
We now give more concrete descriptions of the above colimits. |
1008 |
||
1009 |
In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set, |
|
1010 |
the colimit is |
|
1011 |
\[ |
|
1012 |
\cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) / \sim , |
|
1013 |
\] |
|
1014 |
where $x$ runs through decomposition of $W$, and $\sim$ is the obvious equivalence relation |
|
1015 |
induced by refinement and gluing. |
|
1016 |
If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold, |
|
1017 |
we can take |
|
191
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|
1018 |
\begin{equation*} |
422 | 1019 |
\cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) / K |
191
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1020 |
\end{equation*} |
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1021 |
where $K$ is the vector space spanned by elements $a - g(a)$, with |
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1022 |
$a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
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1023 |
\to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
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1024 |
|
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|
1025 |
In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit |
197 | 1026 |
is more involved. |
142 | 1027 |
%\nn{should probably rewrite this to be compatible with some standard reference} |
191
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|
1028 |
Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
329
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|
1029 |
Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
402 | 1030 |
Define $\cl{\cC}(W)$ as a vector space via |
112 | 1031 |
\[ |
402 | 1032 |
\cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
112 | 1033 |
\] |
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|
1034 |
where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. |
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|
1035 |
(Our homological conventions are non-standard: if a complex $U$ is concentrated in degree $0$, |
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|
1036 |
the complex $U[m]$ is concentrated in degree $m$.) |
422 | 1037 |
\nn{if there is a std convention, should we use it? or are we deliberately bucking tradition?} |
402 | 1038 |
We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
112 | 1039 |
summands plus another term using the differential of the simplicial set of $m$-sequences. |
1040 |
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
|
402 | 1041 |
summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
112 | 1042 |
\[ |
191
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|
1043 |
\bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
112 | 1044 |
\] |
1045 |
where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
|
198 | 1046 |
is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
422 | 1047 |
%\nn{need to say this better} |
1048 |
%\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
|
1049 |
%combine only two balls at a time; for $n=1$ this version will lead to usual definition |
|
1050 |
%of $A_\infty$ category} |
|
108 | 1051 |
|
113 | 1052 |
We will call $m$ the filtration degree of the complex. |
422 | 1053 |
\nn{is there a more standard term for this?} |
113 | 1054 |
We can think of this construction as starting with a disjoint copy of a complex for each |
1055 |
permissible decomposition (filtration degree 0). |
|
1056 |
Then we glue these together with mapping cylinders coming from gluing maps |
|
1057 |
(filtration degree 1). |
|
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1058 |
Then we kill the extra homology we just introduced with mapping |
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|
1059 |
cylinders between the mapping cylinders (filtration degree 2), and so on. |
113 | 1060 |
|
422 | 1061 |
$\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
108 | 1062 |
|
420 | 1063 |
It is easy to see that |
422 | 1064 |
there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps |
108 | 1065 |
comprise a natural transformation of functors. |
1066 |
||
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|
1067 |
\begin{lem} |
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|
1068 |
\label{lem:colim-injective} |
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|
1069 |
Let $W$ be a manifold of dimension less than $n$. Then for each |
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|
1070 |
decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective. |
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|
1071 |
\end{lem} |
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|
1072 |
\begin{proof} |
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|
1073 |
\nn{...} |
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|
1074 |
\end{proof} |
402 | 1075 |
|
108 | 1076 |
\nn{need to finish explaining why we have a system of fields; |
1077 |
define $k$-cat $\cC(\cdot\times W)$} |
|
1078 |
||
1079 |
\subsection{Modules} |
|
95 | 1080 |
|
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|
1081 |
Next we define plain and $A_\infty$ $n$-category modules. |
199 | 1082 |
The definition will be very similar to that of $n$-categories, |
1083 |
but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
|
198 | 1084 |
|
104 | 1085 |
Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary |
102 | 1086 |
in the context of an $m{+}1$-dimensional TQFT. |
1087 |
Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
|
1088 |
This will be explained in more detail as we present the axioms. |
|
1089 |
||
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|
1090 |
Throughout, we fix an $n$-category $\cC$. |
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1091 |
For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. |
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|
1092 |
We state the final axiom, on actions of homeomorphisms, differently in the two cases. |
102 | 1093 |
|
1094 |
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
|
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|
1095 |
$$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
102 | 1096 |
We call $B$ the ball and $N$ the marking. |
1097 |
A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
|
1098 |
restricts to a homeomorphism of markings. |
|
1099 |
||
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|
1100 |
\begin{module-axiom}[Module morphisms] |
102 | 1101 |
{For each $0 \le k \le n$, we have a functor $\cM_k$ from |
1102 |
the category of marked $k$-balls and |
|
1103 |
homeomorphisms to the category of sets and bijections.} |
|
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|
1104 |
\end{module-axiom} |
102 | 1105 |
|
1106 |
(As with $n$-categories, we will usually omit the subscript $k$.) |
|
1107 |
||
423 | 1108 |
For example, let $\cD$ be the TQFT which assigns to a $k$-manifold $N$ the set |
1109 |
of maps from $N$ to $T$ (for $k\le m$), modulo homotopy (and possibly linearized) if $k=m$. |
|
104 | 1110 |
Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary. |
1111 |
Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
|
423 | 1112 |
Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$ |
1113 |
(see Example \ref{ex:maps-with-fiber}). |
|
104 | 1114 |
(The union is along $N\times \bd W$.) |
423 | 1115 |
%(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
1116 |
%the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
|
102 | 1117 |
|
182 | 1118 |
\begin{figure}[!ht] |
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1119 |
$$\mathfig{.8}{ncat/boundary-collar}$$ |
182 | 1120 |
\caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
1121 |
||
103 | 1122 |
Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
1123 |
Call such a thing a {marked $k{-}1$-hemisphere}. |
|
102 | 1124 |
|
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1125 |
\begin{lem} |
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1126 |
\label{lem:hemispheres} |
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|
1127 |
{For each $0 \le k \le n-1$, we have a functor $\cl\cM_k$ from |
104 | 1128 |
the category of marked $k$-hemispheres and |
102 | 1129 |
homeomorphisms to the category of sets and bijections.} |
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|
1130 |
\end{lem} |
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|
1131 |
The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. |
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|
1132 |
We use the same type of colimit construction. |
102 | 1133 |
|
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1134 |
In our example, $\cl\cM(H) = \cD(H\times\bd W \cup \bd H\times W)$. |
104 | 1135 |
|
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|
1136 |
\begin{module-axiom}[Module boundaries (maps)] |
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|
1137 |
{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cl\cM(\bd M)$. |
102 | 1138 |
These maps, for various $M$, comprise a natural transformation of functors.} |
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|
1139 |
\end{module-axiom} |
102 | 1140 |
|
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1141 |
Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
102 | 1142 |
|
1143 |
If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
|
1144 |
then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$ |
|
1145 |
and $c\in \cC(\bd M)$. |
|
1146 |
||
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|
1147 |
\begin{lem}[Boundary from domain and range] |
423 | 1148 |
{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1149 |
$M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere. |
|
104 | 1150 |
Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
424
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|
1151 |
two maps $\bd: \cM(M_i)\to \cl\cM(E)$. |
423 | 1152 |
Then we have an injective map |
102 | 1153 |
\[ |
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|
1154 |
\gl_E : \cM(M_1) \times_{\cl\cM(E)} \cM(M_2) \hookrightarrow \cl\cM(H) |
102 | 1155 |
\] |
1156 |
which is natural with respect to the actions of homeomorphisms.} |
|
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|
1157 |
\end{lem} |
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|
1158 |
Again, this is in exact analogy with Lemma \ref{lem:domain-and-range}. |
102 | 1159 |
|
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|
1160 |
Let $\cl\cM(H)_E$ denote the image of $\gl_E$. |
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|
1161 |
We will refer to elements of $\cl\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$". |
110 | 1162 |
|
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|
1163 |
\begin{lem}[Module to category restrictions] |
103 | 1164 |
{For each marked $k$-hemisphere $H$ there is a restriction map |
424
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|
1165 |
$\cl\cM(H)\to \cC(H)$. |
103 | 1166 |
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
1167 |
These maps comprise a natural transformation of functors.} |
|
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|
1168 |
\end{lem} |
102 | 1169 |
|
103 | 1170 |
Note that combining the various boundary and restriction maps above |
110 | 1171 |
(for both modules and $n$-categories) |
103 | 1172 |
we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$ |
1173 |
a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$. |
|
110 | 1174 |
The subset is the subset of morphisms which are appropriately splittable (transverse to the |
1175 |
cutting submanifolds). |
|
103 | 1176 |
This fact will be used below. |
102 | 1177 |
|
104 | 1178 |
In our example, the various restriction and gluing maps above come from |
1179 |
restricting and gluing maps into $T$. |
|
1180 |
||
1181 |
We require two sorts of composition (gluing) for modules, corresponding to two ways |
|
103 | 1182 |
of splitting a marked $k$-ball into two (marked or plain) $k$-balls. |
119 | 1183 |
(See Figure \ref{zzz3}.) |
103 | 1184 |
|
119 | 1185 |
\begin{figure}[!ht] |
1186 |
\begin{equation*} |
|
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|
1187 |
\mathfig{.4}{ncat/zz3} |
119 | 1188 |
\end{equation*} |
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|
1189 |
\caption{Module composition (top); $n$-category action (bottom).} |
119 | 1190 |
\label{zzz3} |
1191 |
\end{figure} |
|
1192 |
||
1193 |
First, we can compose two module morphisms to get another module morphism. |
|
103 | 1194 |
|
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|
1195 |
\begin{module-axiom}[Module composition] |
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|
1196 |
{Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls (with $0\le k\le n$) |
103 | 1197 |
and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball. |
1198 |
Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere. |
|
1199 |
Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$. |
|
1200 |
We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$. |
|
1201 |
Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps. |
|
1202 |
Then (axiom) we have a map |
|
1203 |
\[ |
|
1204 |
\gl_Y : \cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E \to \cM(M)_E |
|
1205 |
\] |
|
1206 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
1207 |
to the intersection of the boundaries of $M$ and $M_i$. |
|
1208 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
1209 |
(For $k=n$, see below.)} |
|
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|
1210 |
\end{module-axiom} |
119 | 1211 |
|
1212 |
||
103 | 1213 |
Second, we can compose an $n$-category morphism with a module morphism to get another |
1214 |
module morphism. |
|
1215 |
We'll call this the action map to distinguish it from the other kind of composition. |
|
1216 |
||
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|
1217 |
\begin{module-axiom}[$n$-category action] |
103 | 1218 |
{Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$), |
1219 |
$X$ is a plain $k$-ball, |
|
1220 |
and $Y = X\cap M'$ is a $k{-}1$-ball. |
|
1221 |
Let $E = \bd Y$, which is a $k{-}2$-sphere. |
|
1222 |
We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$. |
|
1223 |
Let $\cC(X)_E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps. |
|
1224 |
Then (axiom) we have a map |
|
1225 |
\[ |
|
1226 |
\gl_Y :\cC(X)_E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E |
|
1227 |
\] |
|
1228 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
|
1229 |
to the intersection of the boundaries of $X$ and $M'$. |
|
1230 |
If $k < n$ we require that $\gl_Y$ is injective. |
|
1231 |
(For $k=n$, see below.)} |
|
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|
1232 |
\end{module-axiom} |
103 | 1233 |
|
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|
1234 |
\begin{module-axiom}[Strict associativity] |
423 | 1235 |
The composition and action maps above are strictly associative. |
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|
1236 |
\end{module-axiom} |
103 | 1237 |
|
423 | 1238 |
\nn{should say that this is multifold, not just 3-fold} |
1239 |
||
110 | 1240 |
Note that the above associativity axiom applies to mixtures of module composition, |
1241 |
action maps and $n$-category composition. |
|
119 | 1242 |
See Figure \ref{zzz1b}. |
1243 |
||
1244 |
\begin{figure}[!ht] |
|
1245 |
\begin{equation*} |
|
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|
1246 |
\mathfig{0.49}{ncat/zz0} \mathfig{0.49}{ncat/zz1} |
119 | 1247 |
\end{equation*} |
1248 |
\caption{Two examples of mixed associativity} |
|
1249 |
\label{zzz1b} |
|
1250 |
\end{figure} |
|
1251 |
||
110 | 1252 |
|
1253 |
The above three axioms are equivalent to the following axiom, |
|
103 | 1254 |
which we state in slightly vague form. |
1255 |
\nn{need figure for this} |
|
1256 |
||
1257 |
\xxpar{Module multi-composition:} |
|
1258 |
{Given any decomposition |
|
1259 |
\[ |
|
1260 |
M = X_1 \cup\cdots\cup X_p \cup M_1\cup\cdots\cup M_q |
|
1261 |
\] |
|
1262 |
of a marked $k$-ball $M$ |
|
1263 |
into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a |
|
1264 |
map from an appropriate subset (like a fibered product) |
|
1265 |
of |
|
1266 |
\[ |
|
1267 |
\cC(X_1)\times\cdots\times\cC(X_p) \times \cM(M_1)\times\cdots\times\cM(M_q) |
|
1268 |
\] |
|
1269 |
to $\cM(M)$, |
|
1270 |
and these various multifold composition maps satisfy an |
|
1271 |
operad-type strict associativity condition.} |
|
1272 |
||
423 | 1273 |
The above operad-like structure is analogous to the swiss cheese operad |
1274 |
\cite{MR1718089}. |
|
1275 |
||
1276 |
\medskip |
|
1277 |
||
1278 |
We can define marked pinched products $\pi:E\to M$ of marked balls analogously to the |
|
1279 |
plain ball case. |
|
1280 |
Note that a marked pinched product can be decomposed into either |
|
1281 |
two marked pinched products or a plain pinched product and a marked pinched product. |
|
1282 |
\nn{should give figure} |
|
103 | 1283 |
|
423 | 1284 |
\begin{module-axiom}[Product (identity) morphisms] |
1285 |
For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked |
|
1286 |
$k{+}m$-ball ($m\ge 1$), |
|
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1287 |
there is a map $\pi^*:\cM(M)\to \cM(E)$. |
423 | 1288 |
These maps must satisfy the following conditions. |
1289 |
\begin{enumerate} |
|
1290 |
\item |
|
1291 |
If $\pi:E\to M$ and $\pi':E'\to M'$ are marked pinched products, and |
|
1292 |
if $f:M\to M'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
|
103 | 1293 |
\[ \xymatrix{ |
423 | 1294 |
E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
103 | 1295 |
M \ar[r]^{f} & M' |
1296 |
} \] |
|
423 | 1297 |
commutes, then we have |
1298 |
\[ |
|
1299 |
\pi'^*\circ f = \tilde{f}\circ \pi^*. |
|
1300 |
\] |
|
1301 |
\item |
|
1302 |
Product morphisms are compatible with module composition and module action. |
|
1303 |
Let $\pi:E\to M$, $\pi_1:E_1\to M_1$, and $\pi_2:E_2\to M_2$ |
|
1304 |
be pinched products with $E = E_1\cup E_2$. |
|
1305 |
Let $a\in \cM(M)$, and let $a_i$ denote the restriction of $a$ to $M_i\sub M$. |
|
1306 |
Then |
|
1307 |
\[ |
|
1308 |
\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
|
1309 |
\] |
|
1310 |
Similarly, if $\rho:D\to X$ is a pinched product of plain balls and |
|
1311 |
$E = D\cup E_1$, then |
|
1312 |
\[ |
|
1313 |
\pi^*(a) = \rho^*(a')\bullet \pi_1^*(a_1), |
|
1314 |
\] |
|
1315 |
where $a'$ is the restriction of $a$ to $D$. |
|
1316 |
\item |
|
1317 |
Product morphisms are associative. |
|
1318 |
If $\pi:E\to M$ and $\rho:D\to E$ are marked pinched products then |
|
1319 |
\[ |
|
1320 |
\rho^*\circ\pi^* = (\pi\circ\rho)^* . |
|
1321 |
\] |
|
1322 |
\item |
|
1323 |
Product morphisms are compatible with restriction. |
|
1324 |
If we have a commutative diagram |
|
1325 |
\[ \xymatrix{ |
|
1326 |
D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ |
|
1327 |
Y \ar@{^(->}[r] & M |
|
1328 |
} \] |
|
1329 |
such that $\rho$ and $\pi$ are pinched products, then |
|
1330 |
\[ |
|
1331 |
\res_D\circ\pi^* = \rho^*\circ\res_Y . |
|
1332 |
\] |
|
1333 |
($Y$ could be either a marked or plain ball.) |
|
1334 |
\end{enumerate} |
|
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1335 |
\end{module-axiom} |
103 | 1336 |
|
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1337 |
As in the $n$-category definition, once we have product morphisms we can define |
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|
1338 |
collar maps $\cM(M)\to \cM(M)$. |
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|
1339 |
Note that there are two cases: |
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|
1340 |
the collar could intersect the marking of the marked ball $M$, in which case |
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|
1341 |
we use a product on a morphism of $\cM$; or the collar could be disjoint from the marking, |
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|
1342 |
in which case we use a product on a morphism of $\cC$. |
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1343 |
|
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1344 |
In our example, elements $a$ of $\cM(M)$ maps to $T$, and $\pi^*(a)$ is the pullback of |
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1345 |
$a$ along a map associated to $\pi$. |
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1346 |
|
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1347 |
\medskip |
110 | 1348 |
|
103 | 1349 |
There are two alternatives for the next axiom, according whether we are defining |
1350 |
modules for plain $n$-categories or $A_\infty$ $n$-categories. |
|
1351 |
In the plain case we require |
|
1352 |
||
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1353 |
\begin{module-axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$] |
103 | 1354 |
{Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts |
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1355 |
to the identity on $\bd M$ and is isotopic (rel boundary) to the identity. |
103 | 1356 |
Then $f$ acts trivially on $\cM(M)$.} |
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1357 |
In addition, collar maps act trivially on $\cM(M)$. |
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1358 |
\end{module-axiom} |
103 | 1359 |
|
1360 |
We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense. |
|
1361 |
In other words, if $M = (B, N)$ then we require only that isotopies are fixed |
|
1362 |
on $\bd B \setmin N$. |
|
1363 |
||
1364 |
For $A_\infty$ modules we require |
|
1365 |
||
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1366 |
\addtocounter{module-axiom}{-1} |
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1367 |
\begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act] |
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1368 |
For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes |
103 | 1369 |
\[ |
1370 |
C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) . |
|
1371 |
\] |
|
1372 |
Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$ |
|
1373 |
which fix $\bd M$. |
|
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1374 |
These action maps are required to be associative up to homotopy, |
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1375 |
and also compatible with composition (gluing) in the sense that |
236 | 1376 |
a diagram like the one in Proposition \ref{CHprop} commutes. |
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1377 |
\end{module-axiom} |
103 | 1378 |
|
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1379 |
As with the $n$-category version of the above axiom, we should also have families of collar maps act. |
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1380 |
|
103 | 1381 |
\medskip |
102 | 1382 |
|
104 | 1383 |
Note that the above axioms imply that an $n$-category module has the structure |
1384 |
of an $n{-}1$-category. |
|
1385 |
More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$, |
|
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1386 |
where $X$ is a $k$-ball and in the product $X\times J$ we pinch |
104 | 1387 |
above the non-marked boundary component of $J$. |
200 | 1388 |
(More specifically, we collapse $X\times P$ to a single point, where |
1389 |
$P$ is the non-marked boundary component of $J$.) |
|
104 | 1390 |
Then $\cE$ has the structure of an $n{-}1$-category. |
102 | 1391 |
|
105 | 1392 |
All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds |
1393 |
are oriented or Spin (but not unoriented or $\text{Pin}_\pm$). |
|
1394 |
In this case ($k=1$ and oriented or Spin), there are two types |
|
1395 |
of marked 1-balls, call them left-marked and right-marked, |
|
1396 |
and hence there are two types of modules, call them right modules and left modules. |
|
1397 |
In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
|
1398 |
there is no left/right module distinction. |
|
1399 |
||
130 | 1400 |
\medskip |
1401 |
||
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1402 |
We now give some examples of modules over topological and $A_\infty$ $n$-categories. |
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1403 |
|
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1404 |
\begin{example}[Examples from TQFTs] |
425 | 1405 |
\rm |
1406 |
Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold, |
|
1407 |
and $\cF(W)$ the $j$-category associated to $W$. |
|
1408 |
Let $Y$ be an $(n{-}j{+}1)$-manifold with $\bd Y = W$. |
|
1409 |
Define a $\cF(W)$ module $\cF(Y)$ as follows. |
|
1410 |
If $M = (B, N)$ is a marked $k$-ball with $k<j$ let |
|
1411 |
$\cF(Y)(M)\deq \cF((B\times W) \cup (N\times Y))$. |
|
1412 |
If $M = (B, N)$ is a marked $j$-ball and $c\in \cl{\cF(Y)}(\bd M)$ let |
|
1413 |
$\cF(Y)(M)\deq A_\cF((B\times W) \cup (N\times Y); c)$. |
|
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|
1414 |
\end{example} |
108 | 1415 |
|
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1416 |
\begin{example} |
425 | 1417 |
\rm |
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1418 |
Suppose $S$ is a topological space, with a subspace $T$. |
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1419 |
We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ |
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1420 |
for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs |
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1421 |
$(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all |
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1422 |
such maps modulo homotopies fixed on $\bdy B \setminus N$. |
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|
1423 |
This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. |
420 | 1424 |
\end{example} |
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1425 |
Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and |
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|
1426 |
\ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to |
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1427 |
Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains. |
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|
1428 |
|
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|
1429 |
\subsection{Modules as boundary labels (colimits for decorated manifolds)} |
112 | 1430 |
\label{moddecss} |
108 | 1431 |
|
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1432 |
Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. |
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1433 |
Let $W$ be a $k$-manifold ($k\le n$), |
143 | 1434 |
let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
1435 |
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
|
1436 |
||
1437 |
%Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$), |
|
1438 |
%and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary |
|
1439 |
%component $\bd_i W$ of $W$. |
|
1440 |
%(More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.) |
|
108 | 1441 |
|
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1442 |
We will define a set $\cC(W, \cN)$ using a colimit construction similar to |
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1443 |
the one appearing in \S \ref{ss:ncat_fields} above. |
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|
1444 |
(If $k = n$ and our $n$-categories are enriched, then |
108 | 1445 |
$\cC(W, \cN)$ will have additional structure; see below.) |
1446 |
||
1447 |
Define a permissible decomposition of $W$ to be a decomposition |
|
1448 |
\[ |
|
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|
1449 |
W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) , |
108 | 1450 |
\] |
1451 |
where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
|
1452 |
each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
|
143 | 1453 |
with $M_{ib}\cap Y_i$ being the marking. |
1454 |
(See Figure \ref{mblabel}.) |
|
1455 |
\begin{figure}[!ht]\begin{equation*} |
|
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|
1456 |
\mathfig{.4}{ncat/mblabel} |
143 | 1457 |
\end{equation*}\caption{A permissible decomposition of a manifold |
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|
1458 |
whose boundary components are labeled by $\cC$ modules $\{\cN_i\}$. |
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|
1459 |
Marked balls are shown shaded, plain balls are unshaded.}\label{mblabel}\end{figure} |
108 | 1460 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
1461 |
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
|
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|
1462 |
This defines a partial ordering $\cell(W)$, which we will think of as a category. |
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|
1463 |
(The objects of $\cell(D)$ are permissible decompositions of $W$, and there is a unique |
108 | 1464 |
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
1465 |
||
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|
1466 |
The collection of modules $\cN$ determines |
329
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|
1467 |
a functor $\psi_\cN$ from $\cell(W)$ to the category of sets |
108 | 1468 |
(possibly with additional structure if $k=n$). |
329
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|
1469 |
For a decomposition $x = (X_a, M_{ib})$ in $\cell(W)$, define $\psi_\cN(x)$ to be the subset |
108 | 1470 |
\[ |
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|
1471 |
\psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right) |
108 | 1472 |
\] |
1473 |
such that the restrictions to the various pieces of shared boundaries amongst the |
|
1474 |
$X_a$ and $M_{ib}$ all agree. |
|
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|
1475 |
(That is, the fibered product over the boundary maps.) |
108 | 1476 |
If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
1477 |
via the gluing (composition or action) maps from $\cC$ and the $\cN_i$. |
|
1478 |
||
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|
1479 |
We now define the set $\cC(W, \cN)$ to be the colimit of the functor $\psi_\cN$. |
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|
1480 |
(As usual, if $k=n$ and we are in the $A_\infty$ case, then ``colimit" means |
143 | 1481 |
homotopy colimit.) |
108 | 1482 |
|
143 | 1483 |
If $D$ is an $m$-ball, $0\le m \le n-k$, then we can similarly define |
1484 |
$\cC(D\times W, \cN)$, where in this case $\cN_i$ labels the submanifold |
|
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1485 |
$D\times Y_i \sub \bd(D\times W)$. |
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|
1486 |
It is not hard to see that the assignment $D \mapsto \cC(D\times W, \cN)$ |
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|
1487 |
has the structure of an $n{-}k$-category, which we call $\cT(W, \cN)(D)$. |
144 | 1488 |
|
1489 |
\medskip |
|
1490 |
||
1491 |
||
1492 |
We will use a simple special case of the above |
|
1493 |
construction to define tensor products |
|
1494 |
of modules. |
|
1495 |
Let $\cM_1$ and $\cM_2$ be modules for an $n$-category $\cC$. |
|
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|
1496 |
(If $k=1$ and our manifolds are oriented, then one should be |
144 | 1497 |
a left module and the other a right module.) |
1498 |
Choose a 1-ball $J$, and label the two boundary points of $J$ by $\cM_1$ and $\cM_2$. |
|
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|
1499 |
Define the tensor product $\cM_1 \tensor \cM_2$ to be the |
340
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|
1500 |
$n{-}1$-category $\cT(J, \{\cM_1, \cM_2\})$. |
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|
1501 |
This of course depends (functorially) |
144 | 1502 |
on the choice of 1-ball $J$. |
105 | 1503 |
|
144 | 1504 |
We will define a more general self tensor product (categorified coend) below. |
1505 |
||
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1506 |
|
291 | 1507 |
\subsection{Morphisms of $A_\infty$ $1$-category modules} |
288 | 1508 |
\label{ss:module-morphisms} |
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|
1509 |
|
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|
1510 |
In order to state and prove our version of the higher dimensional Deligne conjecture |
426
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|
1511 |
(\S\ref{sec:deligne}), |
291 | 1512 |
we need to define morphisms of $A_\infty$ $1$-category modules and establish |
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|
1513 |
some of their elementary properties. |
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1514 |
|
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1515 |
To motivate the definitions which follow, consider algebras $A$ and $B$, |
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1516 |
right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction |
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|
1517 |
\begin{eqnarray*} |
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|
1518 |
\hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\ |
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|
1519 |
f &\mapsto& [x \mapsto f(x\ot -)] \\ |
279 | 1520 |
{}[x\ot y \mapsto g(x)(y)] & \mapsfrom & g . |
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|
1521 |
\end{eqnarray*} |
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|
1522 |
If $A$ and $Z_A$ are both the ground field $\k$, this simplifies to |
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|
1523 |
\[ |
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|
1524 |
(X_B\ot {_BY})^* \cong \hom_B(X_B \to (_BY)^*) . |
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|
1525 |
\] |
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|
1526 |
We will establish the analogous isomorphism for a topological $A_\infty$ 1-cat $\cC$ |
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|
1527 |
and modules $\cM_\cC$ and $_\cC\cN$, |
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|
1528 |
\[ |
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|
1529 |
(\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . |
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|
1530 |
\] |
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|
1531 |
|
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|
1532 |
In the next few paragraphs we define the objects appearing in the above equation: |
259
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|
1533 |
$\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally |
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|
1534 |
$\hom_\cC$. |
258
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1535 |
|
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|
1536 |
|
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|
1537 |
\def\olD{{\overline D}} |
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|
1538 |
\def\cbar{{\bar c}} |
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|
1539 |
In the previous subsection we defined a tensor product of $A_\infty$ $n$-category modules |
258
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|
1540 |
for general $n$. |
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|
1541 |
For $n=1$ this definition is a homotopy colimit indexed by subdivisions of a fixed interval $J$ |
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|
1542 |
and their gluings (antirefinements). |
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|
1543 |
(This tensor product depends functorially on the choice of $J$.) |
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|
1544 |
To a subdivision $D$ |
258
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|
1545 |
\[ |
261
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|
1546 |
J = I_1\cup \cdots\cup I_p |
258
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|
1547 |
\] |
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|
1548 |
we associate the chain complex |
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|
1549 |
\[ |
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|
1550 |
\psi(D) = \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) . |
258
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|
1551 |
\] |
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|
1552 |
To each antirefinement we associate a chain map using the composition law of $\cC$ and the |
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|
1553 |
module actions of $\cC$ on $\cM$ and $\cN$. |
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|
1554 |
The underlying graded vector space of the homotopy colimit is |
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|
1555 |
\[ |
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|
1556 |
\bigoplus_l \bigoplus_{\olD} \psi(D_0)[l] , |
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|
1557 |
\] |
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|
1558 |
where $l$ runs through the natural numbers, $\olD = (D_0\to D_1\to\cdots\to D_l)$ |
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|
1559 |
runs through chains of antirefinements of length $l+1$, and $[l]$ denotes a grading shift. |
259
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|
1560 |
We will denote an element of the summand indexed by $\olD$ by |
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|
1561 |
$\olD\ot m\ot\cbar\ot n$, where $m\ot\cbar\ot n \in \psi(D_0)$. |
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|
1562 |
The boundary map is given by |
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|
1563 |
\begin{align*} |
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|
1564 |
\bd(\olD\ot m\ot\cbar\ot n) &= (\bd_0 \olD)\ot \rho(m\ot\cbar\ot n) + (\bd_+ \olD)\ot m\ot\cbar\ot n \; + \\ |
291 | 1565 |
& \qquad + (-1)^l \olD\ot\bd m\ot\cbar\ot n + (-1)^{l+\deg m} \olD\ot m\ot\bd \cbar\ot n + \\ |
1566 |
& \qquad + (-1)^{l+\deg m + \deg \cbar} \olD\ot m\ot \cbar\ot \bd n |
|
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|
1567 |
\end{align*} |
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|
1568 |
where $\bd_+ \olD = \sum_{i>0} (-1)^i (D_0\to \cdots \to \widehat{D_i} \to \cdots \to D_l)$ (those parts of the simplicial |
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|
1569 |
boundary which retain $D_0$), $\bd_0 \olD = (D_1 \to \cdots \to D_l)$, |
259
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|
1570 |
and $\rho$ is the gluing map associated to the antirefinement $D_0\to D_1$. |
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|
1571 |
|
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|
1572 |
$(\cM_\cC\ot {_\cC\cN})^*$ is just the dual chain complex to $\cM_\cC\ot {_\cC\cN}$: |
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|
1573 |
\[ |
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|
1574 |
\prod_l \prod_{\olD} (\psi(D_0)[l])^* , |
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|
1575 |
\] |
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|
1576 |
where $(\psi(D_0)[l])^*$ denotes the linear dual. |
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|
1577 |
The boundary is given by |
291 | 1578 |
\begin{align} |
1579 |
\label{eq:tensor-product-boundary} |
|
1580 |
(-1)^{\deg f +1} (\bd f)(\olD\ot m\ot\cbar\ot n) & = f((\bd_0 \olD)\ot \rho(m\ot\cbar\ot n)) + f((\bd_+ \olD)\ot m\ot\cbar\ot n) + \\ |
|
1581 |
& \qquad + (-1)^{l} f(\olD\ot\bd m\ot\cbar \ot n) + (-1)^{l + \deg m} f(\olD\ot m\ot\bd \cbar \ot n) + \notag \\ |
|
1582 |
& \qquad + (-1)^{l + \deg m + \deg \cbar} f(\olD\ot m\ot\cbar\ot \bd n). \notag |
|
1583 |
\end{align} |
|
259
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|
1584 |
|
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|
1585 |
Next we define the dual module $(_\cC\cN)^*$. |
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|
1586 |
This will depend on a choice of interval $J$, just as the tensor product did. |
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|
1587 |
Recall that $_\cC\cN$ is, among other things, a functor from right-marked intervals |
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|
1588 |
to chain complexes. |
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|
1589 |
Given $J$, we define for each $K\sub J$ which contains the {\it left} endpoint of $J$ |
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|
1590 |
\[ |
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|
1591 |
(_\cC\cN)^*(K) \deq ({_\cC\cN}(J\setmin K))^* , |
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|
1592 |
\] |
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|
1593 |
where $({_\cC\cN}(J\setmin K))^*$ denotes the (linear) dual of the chain complex associated |
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|
1594 |
to the right-marked interval $J\setmin K$. |
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|
1595 |
This extends to a functor from all left-marked intervals (not just those contained in $J$). |
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|
1596 |
\nn{need to say more here; not obvious how homeomorphisms act} |
259
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|
1597 |
It's easy to verify the remaining module axioms. |
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|
1598 |
|
260 | 1599 |
Now we reinterpret $(\cM_\cC\ot {_\cC\cN})^*$ |
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|
1600 |
as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$. |
260 | 1601 |
Let $f\in (\cM_\cC\ot {_\cC\cN})^*$. |
261
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|
1602 |
Let $\olD = (D_0\cdots D_l)$ be a chain of subdivisions with $D_0 = [J = I_1\cup\cdots\cup I_m]$. |
291 | 1603 |
Recall that for any subdivision $J = I_1\cup\cdots\cup I_p$, $(_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) = (_\cC\cN(I_p))^*$. |
261
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|
1604 |
Then for each such $\olD$ we have a degree $l$ map |
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|
1605 |
\begin{eqnarray*} |
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|
1606 |
\cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) &\to& (_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) \\ |
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|
1607 |
m\ot \cbar &\mapsto& [n\mapsto f(\olD\ot m\ot \cbar\ot n)] |
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|
1608 |
\end{eqnarray*} |
260 | 1609 |
|
261
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|
1610 |
We are almost ready to give the definition of morphisms between arbitrary modules |
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|
1611 |
$\cX_\cC$ and $\cY_\cC$. |
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|
1612 |
Note that the rightmost interval $I_m$ does not appear above, except implicitly in $\olD$. |
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|
1613 |
To fix this, we define subdivisions as antirefinements of left-marked intervals. |
261
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|
1614 |
Subdivisions are just the obvious thing, but antirefinements are defined to mimic |
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|
1615 |
the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always |
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|
1616 |
omitted. |
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|
1617 |
More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
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|
1618 |
gluing subintervals together and/or omitting some of the rightmost subintervals. |
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|
1619 |
(See Figure \ref{fig:lmar}.) |
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|
1620 |
\begin{figure}[t]$$ |
381
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|
1621 |
\definecolor{arcolor}{rgb}{.75,.4,.1} |
386 | 1622 |
\begin{tikzpicture}[line width=1pt] |
366
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|
1623 |
\fill (0,0) circle (.1); |
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|
1624 |
\draw (0,0) -- (2,0); |
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|
1625 |
\draw (1,0.1) -- (1,-0.1); |
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|
1626 |
|
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|
1627 |
\draw [->, arcolor] (1,0.25) -- (1,0.75); |
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|
1628 |
|
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|
1629 |
\fill (0,1) circle (.1); |
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|
1630 |
\draw (0,1) -- (2,1); |
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|
1631 |
\end{tikzpicture} |
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|
1632 |
\qquad |
386 | 1633 |
\begin{tikzpicture}[line width=1pt] |
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|
1634 |
\fill (0,0) circle (.1); |
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|
1635 |
\draw (0,0) -- (2,0); |
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|
1636 |
\draw (1,0.1) -- (1,-0.1); |
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|
1637 |
|
381
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|
1638 |
\draw [->, arcolor] (1,0.25) -- (1,0.75); |
366
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|
1639 |
|
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|
1640 |
\fill (0,1) circle (.1); |
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|
1641 |
\draw (0,1) -- (1,1); |
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|
1642 |
\end{tikzpicture} |
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|
1643 |
\qquad |
386 | 1644 |
\begin{tikzpicture}[line width=1pt] |
366
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|
1645 |
\fill (0,0) circle (.1); |
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|
1646 |
\draw (0,0) -- (3,0); |
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|
1647 |
\foreach \x in {0.5, 1.0, 1.25, 1.5, 2.0, 2.5} { |
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|
1648 |
\draw (\x,0.1) -- (\x,-0.1); |
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|
1649 |
} |
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|
1650 |
|
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|
1651 |
\draw [->, arcolor] (1,0.25) -- (1,0.75); |
366
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|
1652 |
|
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|
1653 |
\fill (0,1) circle (.1); |
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|
1654 |
\draw (0,1) -- (2,1); |
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|
1655 |
\foreach \x in {1.0, 1.5} { |
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|
1656 |
\draw (\x,1.1) -- (\x,0.9); |
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|
1657 |
} |
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|
1658 |
|
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|
1659 |
\end{tikzpicture} |
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|
1660 |
$$ |
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|
1661 |
\caption{Antirefinements of left-marked intervals}\label{fig:lmar}\end{figure} |
261
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|
1662 |
|
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|
1663 |
Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
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|
1664 |
The underlying vector space is |
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|
1665 |
\[ |
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|
1666 |
\prod_l \prod_{\olD} \hom[l]\left( |
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|
1667 |
\cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to |
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|
1668 |
\cY(I_1\cup\cdots\cup I_{p-1}) \rule{0pt}{1.1em}\right) , |
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|
1669 |
\] |
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|
1670 |
where, as usual $\olD = (D_0\cdots D_l)$ is a chain of antirefinements |
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|
1671 |
(but now of left-marked intervals) and $D_0$ is the subdivision $I_1\cup\cdots\cup I_{p-1}$. |
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|
1672 |
$\hom[l](- \to -)$ means graded linear maps of degree $l$. |
260 | 1673 |
|
261
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|
1674 |
\nn{small issue (pun intended): |
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|
1675 |
the above is a vector space only if the class of subdivisions is a set, e.g. only if |
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|
1676 |
all of our left-marked intervals are contained in some universal interval (like $J$ above). |
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|
1677 |
perhaps we should give another version of the definition in terms of natural transformations of functors.} |
260 | 1678 |
|
261
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|
1679 |
Abusing notation slightly, we will denote elements of the above space by $g$, with |
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|
1680 |
\[ |
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|
1681 |
\olD\ot x \ot \cbar \mapsto g(\olD\ot x \ot \cbar) \in \cY(I_1\cup\cdots\cup I_{p-1}) . |
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|
1682 |
\] |
340
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|
1683 |
For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, |
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|
1684 |
where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and |
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|
1685 |
$\cbar''$ corresponds to the subintervals |
261
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|
1686 |
which are dropped off the right side. |
386 | 1687 |
(If no such subintervals are dropped, then $\cbar''$ is empty.) |
291 | 1688 |
Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, |
261
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|
1689 |
we have |
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|
1690 |
\begin{eqnarray*} |
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diff
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|
1691 |
(\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\ |
330 | 1692 |
& & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl''(g((\bd_0\olD)\ot \gl'(x\ot\cbar'))\ot\cbar'') . |
261
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|
1693 |
\end{eqnarray*} |
291 | 1694 |
\nn{put in signs, rearrange terms to match order in previous formulas} |
330 | 1695 |
Here $\gl''$ denotes the module action in $\cY_\cC$ |
1696 |
and $\gl'$ denotes the module action in $\cX_\cC$. |
|
261
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|
1697 |
This completes the definition of $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
260 | 1698 |
|
261
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|
1699 |
Note that if $\bd g = 0$, then each |
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|
1700 |
\[ |
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|
1701 |
g(\olD\ot -) : \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to \cY(I_1\cup\cdots\cup I_{p-1}) |
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|
1702 |
\] |
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|
1703 |
constitutes a null homotopy of |
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|
1704 |
$g((\bd \olD)\ot -)$ (where the $g((\bd_0 \olD)\ot -)$ part of $g((\bd \olD)\ot -)$ |
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|
1705 |
should be interpreted as above). |
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|
1706 |
|
410 | 1707 |
Define a {\it strong morphism} |
262
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diff
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|
1708 |
of modules to be a collection of {\it chain} maps |
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|
1709 |
\[ |
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changeset
|
1710 |
h_K : \cX(K)\to \cY(K) |
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diff
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|
1711 |
\] |
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|
1712 |
for each left-marked interval $K$. |
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|
1713 |
These are required to commute with gluing; |
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|
1714 |
for each subdivision $K = I_1\cup\cdots\cup I_q$ the following diagram commutes: |
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diff
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|
1715 |
\[ \xymatrix{ |
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diff
changeset
|
1716 |
\cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_0}\ot \id} |
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|
1717 |
\ar[d]_{\gl} & \cY(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) |
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|
1718 |
\ar[d]^{\gl} \\ |
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|
1719 |
\cX(K) \ar[r]^{h_{K}} & \cY(K) |
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diff
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|
1720 |
} \] |
410 | 1721 |
Given such an $h$ we can construct a morphism $g$, with $\bd g = 0$, as follows. |
262
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|
1722 |
Define $g(\olD\ot - ) = 0$ if the length/degree of $\olD$ is greater than 0. |
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|
1723 |
If $\olD$ consists of the single subdivision $K = I_0\cup\cdots\cup I_q$ then define |
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|
1724 |
\[ |
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|
1725 |
g(\olD\ot x\ot \cbar) \deq h_K(\gl(x\ot\cbar)) . |
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|
1726 |
\] |
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|
1727 |
Trivially, we have $(\bd g)(\olD\ot x \ot \cbar) = 0$ if $\deg(\olD) > 1$. |
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|
1728 |
If $\deg(\olD) = 1$, $(\bd g) = 0$ is equivalent to the fact that $h$ commutes with gluing. |
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|
1729 |
If $\deg(\olD) = 0$, $(\bd g) = 0$ is equivalent to the fact |
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|
1730 |
that each $h_K$ is a chain map. |
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|
1731 |
|
330 | 1732 |
We can think of a general closed element $g\in \hom_\cC(\cX_\cC \to \cY_\cC)$ |
1733 |
as a collection of chain maps which commute with the module action (gluing) up to coherent homotopy. |
|
1734 |
\nn{ideally should give explicit examples of this in low degrees, |
|
1735 |
but skip that for now.} |
|
1736 |
\nn{should also say something about composition of morphisms; well-defined up to homotopy, or maybe |
|
1737 |
should make some arbitrary choice} |
|
262
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|
1738 |
\medskip |
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|
1739 |
|
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|
1740 |
Given $_\cC\cZ$ and $g: \cX_\cC \to \cY_\cC$ with $\bd g = 0$ as above, we next define a chain map |
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|
1741 |
\[ |
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|
1742 |
g\ot\id : \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . |
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|
1743 |
\] |
386 | 1744 |
\nn{...} |
1745 |
More generally, we have a chain map |
|
1746 |
\[ |
|
1747 |
\hom_\cC(\cX_\cC \to \cY_\cC) \ot \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . |
|
1748 |
\] |
|
330 | 1749 |
|
1750 |
\nn{not sure whether to do low degree examples or try to state the general case; ideally both, |
|
1751 |
but maybe just low degrees for now.} |
|
1752 |
||
1753 |
||
1754 |
\nn{...} |
|
262
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|
1755 |
|
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|
1756 |
|
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|
1757 |
|
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|
1758 |
|
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|
1759 |
\medskip |
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|
1760 |
|
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|
1761 |
|
330 | 1762 |
\nn{should we define functors between $n$-cats in a similar way? i.e.\ natural transformations |
1763 |
of the $\cC$ functors which commute with gluing only up to higher morphisms? |
|
1764 |
perhaps worth having both definitions available. |
|
1765 |
certainly the simple kind (strictly commute with gluing) arise in nature.} |
|
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1766 |
|
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1767 |
|
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1768 |
|
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1769 |
|
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1770 |
|
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1771 |
|
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1772 |
|
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1773 |
|
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1774 |
\subsection{The $n{+}1$-category of sphere modules} |
218 | 1775 |
\label{ssec:spherecat} |
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1776 |
|
205 | 1777 |
In this subsection we define an $n{+}1$-category $\cS$ of ``sphere modules" |
327 | 1778 |
whose objects are $n$-categories. |
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1779 |
When $n=2$ |
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|
1780 |
this is closely related to the familiar $2$-category of algebras, bimodules and intertwiners. |
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1781 |
While it is appropriate to call an $S^0$ module a bimodule, |
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|
1782 |
this is much less true for higher dimensional spheres, |
327 | 1783 |
so we prefer the term ``sphere module" for the general case. |
144 | 1784 |
|
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1785 |
The results of this subsection are not needed for the rest of the paper, |
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|
1786 |
so we will skimp on details in a couple of places. We have included this mostly for the sake of comparing our notion of a topological $n$-category to other definitions. |
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|
1787 |
|
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1788 |
For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
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1789 |
|
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1790 |
The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe |
205 | 1791 |
these first. |
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1792 |
The $n{+}1$-dimensional part of $\cS$ consists of intertwiners |
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|
1793 |
of $1$-category modules associated to decorated $n$-balls. |
205 | 1794 |
We will see below that in order for these $n{+}1$-morphisms to satisfy all of |
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|
1795 |
the axioms of an $n{+}1$-category (in particular, duality requirements), we will have to assume |
205 | 1796 |
that our $n$-categories and modules have non-degenerate inner products. |
1797 |
(In other words, we need to assume some extra duality on the $n$-categories and modules.) |
|
1798 |
||
1799 |
\medskip |
|
1800 |
||
1801 |
Our first task is to define an $n$-category $m$-sphere module, for $0\le m \le n-1$. |
|
1802 |
These will be defined in terms of certain classes of marked balls, very similarly |
|
1803 |
to the definition of $n$-category modules above. |
|
1804 |
(This, in turn, is very similar to our definition of $n$-category.) |
|
1805 |
Because of this similarity, we only sketch the definitions below. |
|
1806 |
||
327 | 1807 |
We start with $0$-sphere modules, which also could reasonably be called (categorified) bimodules. |
205 | 1808 |
(For $n=1$ they are precisely bimodules in the usual, uncategorified sense.) |
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|
1809 |
Define a $0$-marked $k$-ball, $1\le k \le n$, to be a pair $(X, M)$ homeomorphic to the standard |
327 | 1810 |
$(B^k, B^{k-1})$. |
209 | 1811 |
See Figure \ref{feb21a}. |
205 | 1812 |
Another way to say this is that $(X, M)$ is homeomorphic to $B^{k-1}\times([-1,1], \{0\})$. |
1813 |
||
209 | 1814 |
\begin{figure}[!ht] |
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1815 |
$$\tikz[baseline,line width=2pt]{\draw[blue] (-2,0)--(2,0); \fill[red] (0,0) circle (0.1);} \qquad \qquad \tikz[baseline,line width=2pt]{\draw[blue][fill=blue!30!white] (0,0) circle (2 and 1); \draw[red] (0,1)--(0,-1);}$$ |
209 | 1816 |
\caption{0-marked 1-ball and 0-marked 2-ball} |
1817 |
\label{feb21a} |
|
1818 |
\end{figure} |
|
1819 |
||
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1820 |
The $0$-marked balls can be cut into smaller balls in various ways. |
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1821 |
We only consider those decompositions in which the smaller balls are either |
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1822 |
$0$-marked (i.e. intersect the $0$-marking of the large ball in a disc) |
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1823 |
or plain (don't intersect the $0$-marking of the large ball). |
327 | 1824 |
We can also take the boundary of a $0$-marked ball, which is $0$-marked sphere. |
205 | 1825 |
|
1826 |
Fix $n$-categories $\cA$ and $\cB$. |
|
327 | 1827 |
These will label the two halves of a $0$-marked $k$-ball. |
205 | 1828 |
|
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|
1829 |
An $n$-category $0$-sphere module $\cM$ over the $n$-categories $\cA$ and $\cB$ is a collection of functors $\cM_k$ from the category |
327 | 1830 |
of $0$-marked $k$-balls, $1\le k \le n$, |
205 | 1831 |
(with the two halves labeled by $\cA$ and $\cB$) to the category of sets. |
1832 |
If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are. |
|
327 | 1833 |
Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have |
205 | 1834 |
morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side) |
1835 |
or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side) |
|
1836 |
or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball). |
|
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1837 |
Corresponding to this decomposition we have a composition (or ``gluing") map |
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|
1838 |
from the product (fibered over the boundary data) of these various sets into $\cM_k(X)$. |
205 | 1839 |
|
1840 |
\medskip |
|
107 | 1841 |
|
327 | 1842 |
Part of the structure of an $n$-category 0-sphere module $\cM$ is captured by saying it is |
206 | 1843 |
a collection $\cD^{ab}$ of $n{-}1$-categories, indexed by pairs $(a, b)$ of objects (0-morphisms) |
1844 |
of $\cA$ and $\cB$. |
|
1845 |
Let $J$ be some standard 0-marked 1-ball (i.e.\ an interval with a marked point in its interior). |
|
1846 |
Given a $j$-ball $X$, $0\le j\le n-1$, we define |
|
1847 |
\[ |
|
1848 |
\cD(X) \deq \cM(X\times J) . |
|
1849 |
\] |
|
1850 |
The product is pinched over the boundary of $J$. |
|
327 | 1851 |
The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$ |
209 | 1852 |
(see Figure \ref{feb21b}). |
206 | 1853 |
These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$. |
107 | 1854 |
|
209 | 1855 |
\begin{figure}[!ht] |
367 | 1856 |
$$ |
1857 |
\begin{tikzpicture}[blue,line width=2pt] |
|
1858 |
\draw (0,1) -- (0,-1) node[below] {$X$}; |
|
1859 |
||
1860 |
\draw (2,0) -- (4,0) node[below] {$J$}; |
|
1861 |
\fill[red] (3,0) circle (0.1); |
|
1862 |
||
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1863 |
\draw[fill=blue!30!white] (6,0) node(a) {} arc (135:90:4) node(top) {} arc (90:45:4) node(b) {} arc (-45:-90:4) node(bottom) {} arc(-90:-135:4); |
367 | 1864 |
\draw[red] (top.center) -- (bottom.center); |
1865 |
\fill (a) circle (0.1) node[left] {\color{green!50!brown} $a$}; |
|
1866 |
\fill (b) circle (0.1) node[right] {\color{green!50!brown} $b$}; |
|
1867 |
||
1868 |
\path (bottom) node[below]{$X \times J$}; |
|
1869 |
||
1870 |
\end{tikzpicture} |
|
1871 |
$$ |
|
209 | 1872 |
\caption{The pinched product $X\times J$} |
1873 |
\label{feb21b} |
|
1874 |
\end{figure} |
|
1875 |
||
206 | 1876 |
More generally, consider an interval with interior marked points, and with the complements |
1877 |
of these points labeled by $n$-categories $\cA_i$ ($0\le i\le l$) and the marked points labeled |
|
1878 |
by $\cA_i$-$\cA_{i+1}$ bimodules $\cM_i$. |
|
209 | 1879 |
(See Figure \ref{feb21c}.) |
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1880 |
To this data we can apply the coend construction as in \S\ref{moddecss} above |
327 | 1881 |
to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category. |
1882 |
This amounts to a definition of taking tensor products of $0$-sphere module over $n$-categories. |
|
205 | 1883 |
|
209 | 1884 |
\begin{figure}[!ht] |
367 | 1885 |
$$ |
1886 |
\begin{tikzpicture}[baseline,line width = 2pt] |
|
1887 |
\draw[blue] (0,0) -- (6,0); |
|
1888 |
\foreach \x/\n in {0.5/0,1.5/1,3/2,4.5/3,5.5/4} { |
|
1889 |
\path (\x,0) node[below] {\color{green!50!brown}$\cA_{\n}$}; |
|
1890 |
} |
|
1891 |
\foreach \x/\n in {1/0,2/1,4/2,5/3} { |
|
1892 |
\fill[red] (\x,0) circle (0.1) node[above] {\color{green!50!brown}$\cM_{\n}$}; |
|
1893 |
} |
|
1894 |
\end{tikzpicture} |
|
1895 |
\qquad |
|
1896 |
\qquad |
|
1897 |
\begin{tikzpicture}[baseline,line width = 2pt] |
|
1898 |
\draw[blue] (0,0) circle (2); |
|
1899 |
\foreach \q/\n in {-45/0,90/1,180/2} { |
|
1900 |
\path (\q:2.4) node {\color{green!50!brown}$\cA_{\n}$}; |
|
1901 |
} |
|
1902 |
\foreach \q/\n in {60/0,120/1,-120/2} { |
|
1903 |
\fill[red] (\q:2) circle (0.1); |
|
1904 |
\path (\q:2.4) node {\color{green!50!brown}$\cM_{\n}$}; |
|
1905 |
} |
|
1906 |
\end{tikzpicture} |
|
1907 |
$$ |
|
209 | 1908 |
\caption{Marked and labeled 1-manifolds} |
1909 |
\label{feb21c} |
|
1910 |
\end{figure} |
|
1911 |
||
206 | 1912 |
We could also similarly mark and label a circle, obtaining an $n{-}1$-category |
1913 |
associated to the marked and labeled circle. |
|
209 | 1914 |
(See Figure \ref{feb21c}.) |
206 | 1915 |
If the circle is divided into two intervals, we can think of this $n{-}1$-category |
327 | 1916 |
as the 2-sided tensor product of the two bimodules associated to the two intervals. |
206 | 1917 |
|
1918 |
\medskip |
|
1919 |
||
1920 |
Next we define $n$-category 1-sphere modules. |
|
1921 |
These are just representations of (modules for) $n{-}1$-categories associated to marked and labeled |
|
1922 |
circles (1-spheres) which we just introduced. |
|
1923 |
||
1924 |
Equivalently, we can define 1-sphere modules in terms of 1-marked $k$-balls, $2\le k\le n$. |
|
1925 |
Fix a marked (and labeled) circle $S$. |
|
209 | 1926 |
Let $C(S)$ denote the cone of $S$, a marked 2-ball (Figure \ref{feb21d}). |
207 | 1927 |
\nn{I need to make up my mind whether marked things are always labeled too. |
1928 |
For the time being, let's say they are.} |
|
1929 |
A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, |
|
1930 |
where $B^j$ is the standard $j$-ball. |
|
399 | 1931 |
A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either |
1932 |
(a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls. |
|
1933 |
(See Figure xxxx.) |
|
207 | 1934 |
We now proceed as in the above module definitions. |
1935 |
||
209 | 1936 |
\begin{figure}[!ht] |
367 | 1937 |
$$ |
1938 |
\begin{tikzpicture}[baseline,line width = 2pt] |
|
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1939 |
\draw[blue][fill=blue!15!white] (0,0) circle (2); |
367 | 1940 |
\fill[red] (0,0) circle (0.1); |
1941 |
\foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} { |
|
1942 |
\draw[red] (0,0) -- (\qm:2); |
|
1943 |
\path (\qa:1) node {\color{green!50!brown} $\cA_\n$}; |
|
1944 |
\path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$}; |
|
1945 |
\draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3); |
|
1946 |
} |
|
1947 |
\end{tikzpicture} |
|
1948 |
$$ |
|
209 | 1949 |
\caption{Cone on a marked circle} |
1950 |
\label{feb21d} |
|
1951 |
\end{figure} |
|
1952 |
||
207 | 1953 |
A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with |
1954 |
\[ |
|
1955 |
\cD(X) \deq \cM(X\times C(S)) . |
|
1956 |
\] |
|
1957 |
The product is pinched over the boundary of $C(S)$. |
|
1958 |
$\cD$ breaks into ``blocks" according to the restriction to the |
|
1959 |
image of $\bd C(S) = S$ in $X\times C(S)$. |
|
1960 |
||
1961 |
More generally, consider a 2-manifold $Y$ |
|
1962 |
(e.g.\ 2-ball or 2-sphere) marked by an embedded 1-complex $K$. |
|
1963 |
The components of $Y\setminus K$ are labeled by $n$-categories, |
|
1964 |
the edges of $K$ are labeled by 0-sphere modules, |
|
1965 |
and the 0-cells of $K$ are labeled by 1-sphere modules. |
|
1966 |
We can now apply the coend construction and obtain an $n{-}2$-category. |
|
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1967 |
If $Y$ has boundary then this $n{-}2$-category is a module for the $n{-}1$-category |
207 | 1968 |
associated to the (marked, labeled) boundary of $Y$. |
1969 |
In particular, if $\bd Y$ is a 1-sphere then we get a 1-sphere module as defined above. |
|
1970 |
||
1971 |
\medskip |
|
1972 |
||
1973 |
It should now be clear how to define $n$-category $m$-sphere modules for $0\le m \le n-1$. |
|
1974 |
For example, there is an $n{-}2$-category associated to a marked, labeled 2-sphere, |
|
208 | 1975 |
and a 2-sphere module is a representation of such an $n{-}2$-category. |
207 | 1976 |
|
1977 |
\medskip |
|
1978 |
||
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1979 |
We can now define the $n$-or-less-dimensional part of our $n{+}1$-category $\cS$. |
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|
1980 |
Choose some collection of $n$-categories, then choose some collections of bimodules between |
207 | 1981 |
these $n$-categories, then choose some collection of 1-sphere modules for the various |
1982 |
possible marked 1-spheres labeled by the $n$-categories and bimodules, and so on. |
|
1983 |
Let $L_i$ denote the collection of $i{-}1$-sphere modules we have chosen. |
|
1984 |
(For convenience, we declare a $(-1)$-sphere module to be an $n$-category.) |
|
1985 |
There is a wide range of possibilities. |
|
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|
1986 |
The set $L_0$ could contain infinitely many $n$-categories or just one. |
207 | 1987 |
For each pair of $n$-categories in $L_0$, $L_1$ could contain no bimodules at all or |
1988 |
it could contain several. |
|
208 | 1989 |
The only requirement is that each $k$-sphere module be a module for a $k$-sphere $n{-}k$-category |
1990 |
constructed out of labels taken from $L_j$ for $j<k$. |
|
1991 |
||
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|
1992 |
We now define $\cS(X)$, for $X$ a ball of dimension at most $n$, to be the set of all |
208 | 1993 |
cell-complexes $K$ embedded in $X$, with the codimension-$j$ parts of $(X, K)$ labeled |
1994 |
by elements of $L_j$. |
|
1995 |
As described above, we can think of each decorated $k$-ball as defining a $k{-}1$-sphere module |
|
1996 |
for the $n{-}k{+}1$-category associated to its decorated boundary. |
|
1997 |
Thus the $k$-morphisms of $\cS$ (for $k\le n$) can be thought |
|
1998 |
of as $n$-category $k{-}1$-sphere modules |
|
1999 |
(generalizations of bimodules). |
|
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2000 |
On the other hand, we can equally well think of the $k$-morphisms as decorations on $k$-balls, |
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2001 |
and from this point of view it is clear that they satisfy all of the axioms of an |
208 | 2002 |
$n{+}1$-category. |
2003 |
(All of the axioms for the less-than-$n{+}1$-dimensional part of an $n{+}1$-category, that is.) |
|
2004 |
||
2005 |
\medskip |
|
2006 |
||
2007 |
Next we define the $n{+}1$-morphisms of $\cS$. |
|
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2008 |
The construction of the 0- through $n$-morphisms was easy and tautological, but the |
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2009 |
$n{+}1$-morphisms will require some effort and combinatorial topology, as well as additional |
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2010 |
duality assumptions on the lower morphisms. These are required because we define the spaces of $n{+}1$-morphisms by making arbitrary choices of incoming and outgoing boundaries for each $n$-ball. The additional duality assumptions are needed to prove independence of our definition form these choices. |
208 | 2011 |
|
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2012 |
Let $X$ be an $n{+}1$-ball, and let $c$ be a decoration of its boundary |
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2013 |
by a cell complex labeled by 0- through $n$-morphisms, as above. |
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2014 |
Choose an $n{-}1$-sphere $E\sub \bd X$ which divides |
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2015 |
$\bd X$ into ``incoming" and ``outgoing" boundary $\bd_-X$ and $\bd_+X$. |
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2016 |
Let $E_c$ denote $E$ decorated by the restriction of $c$ to $E$. |
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2017 |
Recall from above the associated 1-category $\cS(E_c)$. |
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2018 |
We can also have $\cS(E_c)$ modules $\cS(\bd_-X_c)$ and $\cS(\bd_+X_c)$. |
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2019 |
Define |
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2020 |
\[ |
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2021 |
\cS(X; c; E) \deq \hom_{\cS(E_c)}(\cS(\bd_-X_c), \cS(\bd_+X_c)) . |
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2022 |
\] |
208 | 2023 |
|
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2024 |
We will show that if the sphere modules are equipped with a `compatible family of |
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|
2025 |
non-degenerate inner products', then there is a coherent family of isomorphisms |
387
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2026 |
$\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. |
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|
2027 |
This will allow us to define $\cS(X; e)$ independently of the choice of $E$. |
208 | 2028 |
|
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|
2029 |
First we must define ``inner product", ``non-degenerate" and ``compatible". |
387
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2030 |
Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image. |
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|
2031 |
(We assume we are working in the unoriented category.) |
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|
2032 |
Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$ |
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|
2033 |
along their common boundary. |
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|
2034 |
An {\it inner product} on $\cS(Y)$ is a dual vector |
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|
2035 |
\[ |
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|
2036 |
z_Y : \cS(Y\cup\ol{Y}) \to \c. |
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|
2037 |
\] |
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|
2038 |
We will also use the notation |
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|
2039 |
\[ |
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|
2040 |
\langle a, b\rangle \deq z_Y(a\bullet \ol{b}) \in \c . |
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|
2041 |
\] |
390
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|
2042 |
An inner product induces a linear map |
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|
2043 |
\begin{eqnarray*} |
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|
2044 |
\varphi: \cS(Y) &\to& \cS(Y)^* \\ |
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|
2045 |
a &\mapsto& \langle a, \cdot \rangle |
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|
2046 |
\end{eqnarray*} |
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|
2047 |
which satisfies, for all morphisms $e$ of $\cS(\bd Y)$, |
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|
2048 |
\[ |
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|
2049 |
\varphi(ae)(b) = \langle ae, b \rangle = z_Y(a\bullet e\bullet b) = |
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|
2050 |
\langle a, eb \rangle = \varphi(a)(eb) . |
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|
2051 |
\] |
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|
2052 |
In other words, $\varphi$ is a map of $\cS(\bd Y)$ modules. |
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|
2053 |
An inner product is {\it non-degenerate} if $\varphi$ is an isomorphism. |
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|
2054 |
This implies that $\cS(Y; c)$ is finite dimensional for all boundary conditions $c$. |
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|
2055 |
(One can think of these inner products as giving some duality in dimension $n{+}1$; |
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|
2056 |
heretofore we have only assumed duality in dimensions 0 through $n$.) |
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|
2057 |
|
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|
2058 |
Next we define compatibility. |
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|
2059 |
Let $Y = Y_1\cup Y_2$ with $D = Y_1\cap Y_2$. |
398
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|
2060 |
Let $X_1$ and $X_2$ be the two components of $Y\times I$ cut along |
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|
2061 |
$D\times I$, in both cases using the pinched product. |
390
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|
2062 |
(Here we are overloading notation and letting $D$ denote both a decorated and an undecorated |
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|
2063 |
manifold.) |
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|
2064 |
We have $\bd X_i = Y_i \cup \ol{Y}_i \cup (D\times I)$ |
393 | 2065 |
(see Figure \ref{jun23a}). |
2066 |
\begin{figure}[t] |
|
2067 |
\begin{equation*} |
|
2068 |
\mathfig{.6}{tempkw/jun23a} |
|
2069 |
\end{equation*} |
|
2070 |
\caption{$Y\times I$ sliced open} |
|
2071 |
\label{jun23a} |
|
2072 |
\end{figure} |
|
390
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|
2073 |
Given $a_i\in \cS(Y_i)$, $b_i\in \cS(\ol{Y}_i)$ and $v\in\cS(D\times I)$ |
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|
2074 |
which agree on their boundaries, we can evaluate |
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|
2075 |
\[ |
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|
2076 |
z_{Y_i}(a_i\bullet b_i\bullet v) \in \c . |
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|
2077 |
\] |
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|
2078 |
(This requires a choice of homeomorphism $Y_i \cup \ol{Y}_i \cup (D\times I) \cong |
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|
2079 |
Y_i \cup \ol{Y}_i$, but the value of $z_{Y_i}$ is independent of this choice.) |
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|
2080 |
We can think of $z_{Y_i}$ as giving a function |
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|
2081 |
\[ |
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|
2082 |
\psi_i : \cS(Y_i) \ot \cS(\ol{Y}_i) \to \cS(D\times I)^* |
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|
2083 |
\stackrel{\varphi\inv}{\longrightarrow} \cS(D\times I) . |
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|
2084 |
\] |
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|
2085 |
We can now finally define a family of inner products to be {\it compatible} if |
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|
2086 |
for all decompositions $Y = Y_1\cup Y_2$ as above and all $a_i\in \cS(Y_i)$, $b_i\in \cS(\ol{Y}_i)$ |
027bfdae3098
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|
2087 |
we have |
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|
2088 |
\[ |
027bfdae3098
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|
2089 |
z_Y(a_1\bullet a_2\bullet b_1\bullet b_2) = |
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|
2090 |
z_{D\times I}(\psi_1(a_1\ot b_1)\bullet \psi_2(a_2\ot b_2)) . |
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|
2091 |
\] |
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|
2092 |
In other words, the inner product on $Y$ is determined by the inner products on |
027bfdae3098
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|
2093 |
$Y_1$, $Y_2$ and $D\times I$. |
207 | 2094 |
|
392
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|
2095 |
Now we show how to unambiguously identify $\cS(X; c; E)$ and $\cS(X; c; E')$ for any |
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|
2096 |
two choices of $E$ and $E'$. |
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|
2097 |
Consider first the case where $\bd X$ is decomposed as three $n$-balls $A$, $B$ and $C$, |
a7b53f6a339d
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|
2098 |
with $E = \bd(A\cup B)$ and $E' = \bd A$. |
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|
2099 |
We must provide an isomorphism between $\cS(X; c; E) = \hom(\cS(C), \cS(A\cup B))$ |
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|
2100 |
and $\cS(X; c; E') = \hom(\cS(C\cup \ol{B}), \cS(A))$. |
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|
2101 |
Let $D = B\cap A$. |
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|
2102 |
Then as above we can construct a map |
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|
2103 |
\[ |
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|
2104 |
\psi: \cS(B)\ot\cS(\ol{B}) \to \cS(D\times I) . |
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|
2105 |
\] |
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|
2106 |
Given $f\in \hom(\cS(C), \cS(A\cup B))$ we define $f'\in \hom(\cS(C\cup \ol{B}), \cS(A))$ |
a7b53f6a339d
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|
2107 |
to be the composition |
a7b53f6a339d
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changeset
|
2108 |
\[ |
a7b53f6a339d
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|
2109 |
\cS(C\cup \ol{B}) \stackrel{f\ot\id}{\longrightarrow} |
a7b53f6a339d
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|
2110 |
\cS(A\cup B\cup \ol{B}) \stackrel{\id\ot\psi}{\longrightarrow} |
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|
2111 |
\cS(A\cup(D\times I)) \stackrel{\cong}{\longrightarrow} \cS(A) . |
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|
2112 |
\] |
393 | 2113 |
(See Figure \ref{jun23b}.) |
2114 |
\begin{figure}[t] |
|
2115 |
\begin{equation*} |
|
2116 |
\mathfig{.5}{tempkw/jun23b} |
|
2117 |
\end{equation*} |
|
2118 |
\caption{Moving $B$ from top to bottom} |
|
2119 |
\label{jun23b} |
|
2120 |
\end{figure} |
|
392
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|
2121 |
Let $D' = B\cap C$. |
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|
2122 |
Using the inner products there is an adjoint map |
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|
2123 |
\[ |
a7b53f6a339d
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changeset
|
2124 |
\psi^\dagger: \cS(D'\times I) \to \cS(\ol{B})\ot\cS(B) . |
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|
2125 |
\] |
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|
2126 |
Given $f'\in \hom(\cS(C\cup \ol{B}), \cS(A))$ we define $f\in \hom(\cS(C), \cS(A\cup B))$ |
a7b53f6a339d
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|
2127 |
to be the composition |
a7b53f6a339d
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changeset
|
2128 |
\[ |
a7b53f6a339d
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changeset
|
2129 |
\cS(C) \stackrel{\cong}{\longrightarrow} |
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changeset
|
2130 |
\cS(C\cup(D'\times I)) \stackrel{\id\ot\psi^\dagger}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2131 |
\cS(C\cup \ol{B}\cup B) \stackrel{f'\ot\id}{\longrightarrow} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2132 |
\cS(A\cup B) . |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2133 |
\] |
393 | 2134 |
(See Figure \ref{jun23c}.) |
2135 |
\begin{figure}[t] |
|
2136 |
\begin{equation*} |
|
2137 |
\mathfig{.5}{tempkw/jun23c} |
|
2138 |
\end{equation*} |
|
2139 |
\caption{Moving $B$ from bottom to top} |
|
2140 |
\label{jun23c} |
|
2141 |
\end{figure} |
|
2142 |
Let $D' = B\cap C$. |
|
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2143 |
It is not hard too show that the above two maps are mutually inverse. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2144 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2145 |
\begin{lem} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2146 |
Any two choices of $E$ and $E'$ are related by a series of modifications as above. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2147 |
\end{lem} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2148 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2149 |
\begin{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2150 |
(Sketch) |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2151 |
$E$ and $E'$ are isotopic, and any isotopy is |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2152 |
homotopic to a composition of small isotopies which are either |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2153 |
(a) supported away from $E$, or (b) modify $E$ in the simple manner described above. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2154 |
\end{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2155 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2156 |
It follows from the lemma that we can construct an isomorphism |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2157 |
between $\cS(X; c; E)$ and $\cS(X; c; E')$ for any pair $E$, $E'$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2158 |
This construction involves on a choice of simple ``moves" (as above) to transform |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2159 |
$E$ to $E'$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2160 |
We must now show that the isomorphism does not depend on this choice. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2161 |
We will show below that it suffice to check two ``movie moves". |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2162 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2163 |
The first movie move is to push $E$ across an $n$-ball $B$ as above, then push it back. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2164 |
The result is equivalent to doing nothing. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2165 |
As we remarked above, the isomorphisms corresponding to these two pushes are mutually |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2166 |
inverse, so we have invariance under this movie move. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2167 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2168 |
The second movie move replaces to successive pushes in the same direction, |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2169 |
across $B_1$ and $B_2$, say, with a single push across $B_1\cup B_2$. |
393 | 2170 |
(See Figure \ref{jun23d}.) |
2171 |
\begin{figure}[t] |
|
2172 |
\begin{equation*} |
|
2173 |
\mathfig{.9}{tempkw/jun23d} |
|
2174 |
\end{equation*} |
|
2175 |
\caption{A movie move} |
|
2176 |
\label{jun23d} |
|
2177 |
\end{figure} |
|
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2178 |
Invariance under this movie move follows from the compatibility of the inner |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2179 |
product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$. |
411
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
410
diff
changeset
|
2180 |
\nn{should also say something about locality/distant-commutativity} |
392
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2181 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2182 |
If $n\ge 2$, these two movie move suffice: |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2183 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2184 |
\begin{lem} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2185 |
Assume $n\ge 2$ and fix $E$ and $E'$ as above. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2186 |
The any two sequences of elementary moves connecting $E$ to $E'$ |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2187 |
are related by a sequence of the two movie moves defined above. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2188 |
\end{lem} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2189 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2190 |
\begin{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2191 |
(Sketch) |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2192 |
Consider a two parameter family of diffeomorphisms (one parameter family of isotopies) |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2193 |
of $\bd X$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2194 |
Up to homotopy, |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2195 |
such a family is homotopic to a family which can be decomposed |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2196 |
into small families which are either |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2197 |
(a) supported away from $E$, |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2198 |
(b) have boundaries corresponding to the two movie moves above. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2199 |
Finally, observe that the space of $E$'s is simply connected. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2200 |
(This fails for $n=1$.) |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2201 |
\end{proof} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2202 |
|
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2203 |
For $n=1$ we have to check an additional ``global" relations corresponding to |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2204 |
rotating the 0-sphere $E$ around the 1-sphere $\bd X$. |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2205 |
\nn{should check this global move, or maybe cite Frobenius reciprocity result} |
a7b53f6a339d
finished def of sphere module n+1-cat
Kevin Walker <kevin@canyon23.net>
parents:
390
diff
changeset
|
2206 |
|
207 | 2207 |
\nn{...} |
101 | 2208 |
|
2209 |
\medskip |
|
2210 |
\hrule |
|
2211 |
\medskip |
|
2212 |
||
95 | 2213 |
\nn{to be continued...} |
101 | 2214 |
\medskip |
98 | 2215 |
|
2216 |
||
208 | 2217 |
|
2218 |
||
2219 |
||
2220 |
||
98 | 2221 |
Stuff that remains to be done (either below or in an appendix or in a separate section or in |
2222 |
a separate paper): |
|
2223 |
\begin{itemize} |
|
207 | 2224 |
\item discuss Morita equivalence |
139 | 2225 |
\item functors |
98 | 2226 |
\end{itemize} |
2227 |
||
204 | 2228 |